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Unit 1: Mathematical operations on whole Numbers up to 100,000
Key Unity competences
By the end of this unit, a learner should be able to write, compare and calculate whole numbers up to 100,000.Attitude and Values
• develop personal confidence in the use of numbers.
• appreciate the importance of addition, subtraction, division and multiplication of numbers in real life.
• appreciate the need for manipulating numbers.The Number System
A number is a mathematical way of representing how many, how far, how long or how much a quantity is.
A number (numeral) system or system of numeration is a writing system for expressing numbers using digits or other symbols in a consistent manner.Three of the most commonly used number systems are:
1. The Hindu-Arabic Numeral System.
2. The Roman Numeral System.
3. The Chinese Numeral System.The Hindu-Arabic Numeral System
This numeration system uses ten digits; 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. These digits can be used to form any number. For example, the digits 3, 9, 7 can be used to form numbers 397, 379, 973, 937, 793 and 739.
Natural Numbers (Counting Numbers)
When a child is learning to count, he/she says “one, two, three, four... .” This set of numbers is called natural numbers or counting numbers.
Counting (natural) numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9 ...
Whole numbers
Whole numbers is a set of counting (natural) numbers including zero.
Whole numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ...
Forming numerals from digits
The digits 0 to 9 are used to form different numbers depending on how you arrange the digits.
Example 1.1
Write down any 2-digit number formed by the digits 4, 6, 7, 0, 1.
Solution
Pick any two digits given above.
For example
First number = 46
Second number = 67Example 1.2
Give the smallest number or numeral that can be obtained from the digits 8, 9,6,1.
Solution
First, arrange the digits in ascending order (from smallest to biggest).
In ascending order we get 1, 6, 8, 9.
Therefore, the smallest number is 1 689.
Other numbers are 40, 10, 70, 71, 76, 61, etc.Example 1.3
Find the biggest number that can be formed from the digits 1, 3, 5, 9, 4?
Solution
Arrange the digits in descending order (from biggest to smallest).
In descending order, we get 9, 5, 4, 3, 1.
Therefore, the biggest number formed is 95 431.Activity 1.1
In this class activity, you will form numbers from given digits.
(i) Arrange five blank flash cards provided on your table.
(ii) Write the digits 5, 2, 4, 1 and 3 on each flash card.
(iii) Form a 5-digit number starting with 5.
(iv) Form a 5-digit number starting with 2.
(v) Form a 5-digit number starting with 4.
(vi) Form a 5-digit number starting with 1.
(vii) Form a 5-digit number starting with 3.
(viii) Arrange the formed numbers in ascending order.
Exercise 1.1
0,1, 2, 3, 4, 5, 6, 7, 8, 9
You can use a digit only once while answering the questions below:
(a) Use the digits above to form all 3-digit numbers beginning with 3 and ending with 9. {An example is 329}.
(b) What is the largest 2-digit number that can be formed from these digits?
(c) What is the smallest 2-digit number that can be formed from these digits?
Reading and writing numbers in words and in figures
Use one zero ‘0’ to write a ten, two zeroes ‘00’ to write a hundred, three zeroes ‘000’ to write a thousand. For example:10 = Ten.
30 = Thirty.
100 = One hundred.
370 = Three hundred seventy.
1 000 = One thousand.
8 900 = Eight thousand nine hundred.
50 000 = Fifty thousand.
99 990 = Ninety-nine thousand nine hundred ninety.
100 000 = One hundred thousand.
In this class activity, you will play a game of matching numbers.
(i) Divide yourselves into 2 groups: Group A and B.
(ii) Arrange yourselves in a straight line and sit in 2 groups by facing each other. The pupil seated opposite to you is your opponent.
(iii) Write one large number in words and one large number in figures on the two manila cards.(iv) Take turns in playing the game by showing your opponent the numbers you have written on the cards. He/she should say the number aloud or write the number on the blackboard. The teacher together with the rest of the pupils will act as judges.
(v) Correct answer = 3 marks and wrong answer = 0 marks. Add up the marks for your group and see which group wins.
Example 1.4
Write the following numbers in words:
(a) 97 642
(b) 10 002
(c) 999
(d) 90 001
(e) 1 704Solution
(a) 97 642 = Ninety-seven thousand six hundred forty-two.
(b) 10 002 = Ten thousand two.
(c) 999 = Nine hundred ninety-nine.
(d) 90 001 = Ninety thousand one.
(e) 1 704 = One thousand seven hundred four.Example 1.5
Write the following in figures:
(a) Ninety-nine.
(b) One hundred ninety-nine.
(c) Seven hundred five.
(d) Six hundred fifty.
(e) Twelve thousand five hundred.
(f) Eighty thousand.
(g) Forty thousand one.One hundred thousand.
(i) Nine hundred ninety-seven.
(j) Eleven thousand one hundred eleven.Solution
1.21. Read and write the following numbers in words:
(a) 111 (b) 1 011 (c) 54 700
(d) 1 000 (e) 78 999 (f) 15 0002
Write the following numbers in figures.
(a) Nineteen (b) Ninety (c) Ninety-nine (d) Nine hundred ninety-nine.
(e) Five hundred one. (f) Three thousand one.
Place Values of Whole Numbers
Place values help us to show the position of a given digit in a number.
Example 1.6
Find the value of each digit in 79 846.
Example 1.7
Use the table below to work out the value of each digit in the number 43 275.
Solution
Get an abacus frame
(i) Place 3 beads on the ten thousand’s spike, 4 on the thousand’s spike, 2 on the hundreds spike, 7 on the ten’s spike and 1 on the ones spike.
(ii) Read the number you have formed.
Write down the place values of the underlined digits in the given numbers:
Expanding whole numbers using place values
Example 1.8
Expand the following numbers:
(a) 82 645 (b) 98 067
Solution
(a) 82 645 = 80 000 + 2 000 + 600 + 40 + 5
(b) 98 067 = 90 000 + 8 000 + 60 + 7
Writing expanded numbers in short formExample 1.9
Write the following expanded numbers in short form:
(a) 10 + 9 (b) 30 000 + 5 000
(c) 30 000 + 400 + 40 + 4 (d) 90 000 + 1
Solution(a) 10 + 9 (b) 30 000 + 5 000
(c) 30 000 + 400 + 40 + 4 (d) 90 000 + 1
• Make cards like the ones given below.
• Pair the cards which have same value.
Write the following numbers in expanded form.
(a) 4 624 (b) 984 (c) 44 484
(d) 93 428 (e) 76 709
2. Write the following in short form.
(a) (7 × 1 000) + (3 × 100) + (8 × 10) + (3 × 1)
(b) (9 × 10 000) + (8 × 1 000) + (3 × 100) + (7 × 10) + (6 × 1)
(c) (3 × 10 000) + (9 × 1 000) + (8 × 100) + (6 × 10) + (7 × 1)
Comparing Numbers
When comparing two numbers, we count the number of digits in each number.
• The number with more number of digits is greater than the one with few digits.
• The number with fewer digits is smaller than the other.
• If the two numbers have equal number of digits, then we can consider the value of each digit. The number which has greater value is greater than the other.
The following notations are used while comparing numbers:
< denotes ‘less than’
> denotes ‘greater than’ and
= denotes ‘equal to’.
In this class activity, you will write down the biggest number you know.
(i) Form yourselves in groups of 5 learners each.
(ii) Write down the greatest number you know on a flash card.
(iii) Compare the number you have written with the numbers your classmates have written.Example 1.10
Use the symbol <, >, or = to compare the following numbers and items:
(a) 4 thousand and 4 × 1000 (b) 10 000 and 1 000
(c) 500 Frw and 550 Frw (d) 400 bottles and 400 bottlesSolution
(a) 4 thousand is equal to 4 × 1000, because 4 thousand = 4 × 1000
= 4000.
(b) 10 000 > 1 000 because 10 000 has more digits than 1 000.(c) 500 and 550 have the same number of digits. The first digit in both numbers is 5. So, we now compare the next digits which are 0 and 5. Since 5 > 0, therefore, 500 Frw < 550 Frw.
(d) 400 bottles = 400 bottles. The two numbers are exactly the same so they are equal.Think!!!
There is a three-digit number. The second digit is four times as big as the third digit, while the first digit is three less than the second digit. What is the number?
1. Use the correct sign <, > or =
Addition of Whole Numbers
When adding numbers, we arrange the numbers so that the ones are on the same vertical line, the tens are also on the same vertical line and so on.
Addition of Whole Numbers Without Carrying
Example 1.11
SolutionT. TH
TH
H
T
O
3
4
9
0
7
5
4
0
9
1
8
8
9
9
8
2. Add 375 and 8 201.
Solution
Numbers can be easily added by using an addition grid as shown below:
1. Add the following numbers and write your answers in words:
2. Add the following numbers together:
(a) 230 and 230 (b) 21 650 and 32 146
(c) 5 and 21 454 (d) 160 and 610
3. I have three thousand Rwandan Francs and my brother has seven
thousand five hundred Rwandan Francs. How much do I and my
brother have altogether
4. Town A has 242 cars and Town B has 424 cars. How many cars are
there in the two towns?Addition with Carrying
When the sum of two or more numbers is more than 9, we carry as we do the addition as shown in the example given on the next page:
Example 1.12
Add 4 999 and 8 294.
Solution
• Add downwards starting from the place of ones on your right.
• A number where the sum above is a two-digit number, write the ones and carry the tens to the next digit to the left.
• So, 4 999 + 8 294 = 13 293.
1. Add the following numbers together:
(a) 53 250 and 32 488 (b) 738 and 262
(c) 1 354 and 9 999 (d) 999 and 111
(e) 999 src="https://elearning.reb.rw/pluginfile.php/5332/mod_label/intro/Capture%20%282%29.PNG">In this class activity, you will form numbers from given digits.
(i) Arrange five blank flash cards provided on your table.
(ii) Write the digits 5, 2, 4, 1 and 3 on each flash card.
(iii) Form a 5-digit number starting with 5.
(iv) Form a 5-digit number starting with 2.
(v) Form a 5-digit number starting with 4.
(vi) Form a 5-digit number starting with 1.
(vii) Form a 5-digit number starting with 3.
(viii) Arrange the formed numbers in ascending and 1 1102. The data below shows the monthly amount of rainfall received by a certain country for a whole year;
(a) How much total rainfall was received between November and December?
(b) How much rainfall was received in the first three months?(c) Write down in words, the amount of rainfall received in the month of May.
(d) Calculate the sum of the least and greatest amount of rainfall during the year.
Think!!!
Using only addition, how do you add eight 8s and get the number 1 000?
Subtraction
Subtraction without Borrowing
Just like we did with addition of numbers, we arrange numbers so that the Ones, Tens, ... etc. are all aligned.
Example 1.13
Evaluate: 6 989 – 3 453
Solution
Subtraction with Borrowing
Example 1.14
Evaluate: 2 573 – 1 395
• Arrange the two numbers as we did for addition.
• In ones, 3 – 5 is not possible because 3 is less than 5. We then borrow 1 from the next digit under the tens.
• So, we borrow 1 from 7 tens and remain with 6. The one we have borrowed becomes 10 and 10 + 3 = 13.
• Now, we subtract 5 from 13 to get 8.
• Now, in tens 6 – 9 is also impossible. So, we borrow 1 from 5 hundreds, so, 16 – 9 = 7.
• In the hundreds column, we have 4 – 3 = 1 and in the thousands column, we have 2 – 1 = 1.
• So, 2 573 – 1 395 = 1 178.
Example 1.16
Eina has 73 567 Frw in her account and Awiza has 89 504 Frw in his account. What is the difference between the money in Eina's account and in Awiza's account.
Solution
• Put 256 bean seeds in a tin box.
• Remove 75 bean seeds from the tin box.
• Count the number of seeds which remained in the tin.
• How many seeds are left?
1. Evaluate:
(a) 15 789 – 11 000 (b) 86 786 – 63 524
(c) 99 999 – 29 999 (d) 863 – 489
(e) 45 567 – 12 540 (f) 48 487 – 32 450
(g) 32 450 – 16 360
(i) 65 009 – 222 (j) 90 000 – 1
2. A shirt costs 3 000 Frw and another shirt costs 1999 Frw. What is the difference in cost of two shirts?
3. I had 304 eggs in a box, out of which 70 got broken. How many eggs were not broken?
4. One metre of cloth costs 1 250 Frw and another cloth of 1 metre costs 2600 Frw. How much more does the another cloth cost?
5. Kayongo went to the market with 550 Frw and bought mangoes. He was left with a balance of 120 Frw. How much did he use to buy the mangoes?
6. By how much is 67 015 more than 67 010?
7. The distance between London City in the U.K and New York City in USA is 5 546 km. A plane travelling from London to New York has covered a distance of 4 509 km. What distance has to be covered by the plane to reach New York?
8. A district had a population of 30 845 in a national census. Of these, 19 678 were females. How many males were in the district?
9. The following people had the given amounts of money on their bank accounts.
Matsiko
Peter
Lina
Mwiza
Maniraho
24 672
53 765
80 000
87 450
23 908
(a) What is the difference between the money in Matsiko’s account
and in Lina’s account?
(b) What is the difference of money in Mwiza’s and Peter’s accounts?
10. A town has 98 500 buildings. 70 005 of these buildings are shops. Find the number of other types of buildings in the town.Multiplication of whole numbers
The Multiplication Table
Example 1.17
Evaluate:
1. 23 × 2 2. 45 × 7
3. 21 × 13 4. 38 × 85
5. 782 × 3 6. 476 × 25
7. 450 × 25 8. 620 × 25
Solution 1. 23 × 2
Multiplication of whole numbers by 10, 100, 1 000, 10 000.
When multiplying a whole number by:
• 10, we add a zero (0) at the right hand side of end of the last digit of the number.
• 100, we add two zeroes at the right hand side of the last digit of the number.
• 1 000, we add three zeroes at the right hand side of the last digit of the number.
• 10 000, we add four zeroes at the right hand side of the last digit of the number.
This is shown in the cases below:
(a) 27 × 10 = 270
(b) 4 × 10 = 40
(c) 2 376 × 10 = 23 760
(d) 24 × 100 = 2 400
(e) 457 × 100 = 45 700
(f) 34 × 1 000 = 34 000
(g) 6 × 10 000 = 60 000
Quick Multiplication by 5:
In order to multiply a whole number by 5, you have to multiply it by 10, then divide the product by 2.
Therefore, 280 × 5 = (280 × 10) ÷ 2
= 2800 ÷ 2 = 1400
Quick Multiplication by 100 and 1000:
In order to multiply a whole number by 100, we put two zeros on the right hand side of that number. Therefore, 82 × 100 = 8200.
In order to multiply a whole number by 1000, we put three zeros on the right hand side of that number.
Therefore, 46 × 1000 = 46000.
Example 1.18
A square floor of classroom is completely covered with tiles. Length of square room is 26 m. Also number of tiles along its each boundary is 26. How many tiles are there in the class room.
Solution
Number of tiles along one boundary (length) = 26
Number of tiles along another boundary (width) = 26
Total titles = 26 × 26
1. Complete the multiplication table below and learn them by heart.
2. Complete the table below.
3. Work out:
(a) 164 × 6 (b) 40 × 9
(c) 78 × 7 (d) 217 × 5
(e) 450 × 5 (f) 897 × 5
(g) 575 × 3
(i) 999 × 9 (j) 35 × 21
(k) 34 × 12 (l) 56 × 23
(m) 75 × 25
55 × 60
(o) 39 × 13
1. A rectangular floor is completely covered with tiles. There are 26 tiles along its length and 15 tiles along its width. How many tiles are there altogether in the room?2. There are 10 rows of students at a school assembly. Each row has 28 students. How many students are there altogether?
3. Each box contains 500 nails. How many nails are contained in 80 boxes?
4. Kigali Chalk Factory produces 90 cartons of chalk in a day. Each carton contains 36 boxes of chalk. How many boxes of chalk does the factory produce per day?
5. The table below shows part of Mr. and Mrs. Bitega’s wedding budget. The cost of the items is in dollars.
Item
Price per unit
Quantity needed
Total cost for item
Chicken
$ 40
10
Soda
$ 65
15 crates
Wedding cake
$ 305
11 piece
Wedding gown
$ 850
2
Shoes
$ 370
2 pairs
Complete the table and find the total cost for the different items.
Division
Division without A Remainder
Example 1.19
Work out the following:
(a) 48 ÷ 2 (b) 245 ÷ 5
(c) 90 ÷ 5 (d) 2 170 ÷ 7
(e) 54 000 ÷ 45 (f) 108 ÷ 6
Solution
Example 1.20
Divide 5000 pens equally among 5 children.
Solution
Divide 5000 by 5
1. Work out the following:
(a) 14 ÷ 7 (b) 180 ÷ 4
(c) 330 ÷ 11 (d) 900 ÷ 6
(e) 840 ÷ 10 (f) 2 718 ÷ 6
(g) 2 247 ÷ 7
(i) 4 077 ÷ 9 (j) 1 242 ÷ 54
(k) 1 380 ÷ 60 (l) 9 450 ÷ 15 2.
2. Answer the following questions:
(a) There are 340 bags of cement to be loaded off from a lorry by 5 men working at the same rate. How many bags will each man off the load?
(b) Divide 450 sweets equally among 10 pupils.
(c) The District Agriculture Officer plans to distribute 1 080 Friesian cows to 10 farmers’ groups. How many cows will each farmers’ group get?
(d) A butcher supplies 10 kg of meat to each school. If he has 1 000 kg of meat, how many schools does he supply?
Division with a Remainder
Example 1.21
Evaluate the following:
(a) 25 ÷ 4 (b) 398 ÷ 5 (c) 1 478 ÷ 12
Solution
(a) 25 ÷ 4 =
Assessment exercise
1. What is the largest 5-digit number that can be formed from the digits 0, 3, 2, 4, 1, 5? 2. Write the following numbers in figures: (a) Twenty-four thousand seven hundred seven. (b) One hundred twenty-four thousand seven hundred seventy. (c) Thirty-four thousand seven hundred seventy-seven. 3. Write the following in short form. (a) (4 × 1 000) + (2 × 100) + (7 × 10) + (3 × 1) (b) (5 × 10 000) + (5 × 100) + (3 × 10) + (2 × 1) 4. In a school with 888 pupils, three girls were voted for the post of head girl of the school. The number of votes obtained by each girl is given below:
Girl
Number of votes
Umuhoza Bianca
323
Ineza Rebecca
233
Uwimana Christa
332
Which girl obtained the greatest number of votes?
5. On a certain school trip to Akagera National Game Park, we counted 123 zebras, 10 buffaloes, 21 wild pigs and 14 giraffes.(a) How many giraffes and zebras did we count?
(b) How many wild animals did we count altogether in the park? 6. Fill the missing digits in the addition grids below:
7. A shopkeeper in Kimironko main market in Kigali had the following daily records of sales:
(a) What was the value of goods sold on Monday?
(b) What was the value of goods sold on Thursday and Friday?
(c) How much did the shopkeeper get from Monday to Wednesday?
8. A school water tank holds 100 000 litres of water. P5 pupils use 12 500 litres and P6 pupils use 67 500 litres. How much water remains in the tank?
9. By how much is 67 999 greater than 45 908? 10.
Work out:
(a) 217 × 11 (b) 234 × 90 (c) 805 × 30
(d) 565 × 20 (e) 680 × 26 (f) 615 × 10
(g) 575 × 33
(j) 895 × 100 (k) 65 × 456
11. Each tray of eggs contain 30 eggs. How many eggs are in 22 trays?
13. A father had 23 sweets and decided to give the sweets to his four children equally.
(a) How many sweets did each child get?
(b) How many sweets did he remain with?
14. A class has 67 pupils. A teacher wants to form groups of 4 pupils each. (a) How many groups were formed in the class?
(b) How many students remain without a group?
Think!
Which 3 numbers have the same answer whether they are added or multiplied together?
Key Unit Competence
A learner should be able to solve problems related to comparing, ordering and finding distance between negative and positive integers. Attitudes and values Appreciate the importance of using positive and negative numbers in practical contexts. Integer An integer is a number with no fractional part. Integer include all the whole numbers along with negative numbers. We represent the integers on a number line as follows:
A set of integers includes:
1. All counting numbers (1, 2, 3, 4, 5, 6, 7, ………. 100, ..)
2. Zero {0}
3. And the negatives of all the counting numbers (–1,–2,–3,–4,–5….)
Therefore, the set of integers is (……, –3, –2, –1, 0, +1, +2, +3, …..)
Integers are made up of negative numbers, zero and positive numbers.
‘Zero’ is neither positive nor negative.
In this class activity, you will discuss positive and negative numbers with your group members.
You should discuss positive and negative numbers in relation to the following:
(i) Falling into a hole and climbing a tree.
(ii) Profit and loss in business.
(iii) Sinking into water and rising above earth.
(iv) Promotion and demotion at work place.
Explaining Integers
• If I have no money at all, then I can say that I
have ‘0’ money.
• If I have been given some 50 Frw, then I can say I have some money.
• If I have lost 100, then I can say that I have a shortage of 100 Frw.
• If a stone is dropped into a pit which is 10 metres
deep, then from the ground level the stone is at
–10 m from the earth surface.
• A plane flying above our school at a height of
500 m is +500 m above the school.
Example 2.1
Study the diagram below and use integers to estimate the position of the
following objects from the earth surface.
Solution
(a) The position of the plane is +40 m.
(b) The position of the balloon is +35 m.
(c) The position of the ball is +10 m.
(d) The position of the ground is 0 m.
(e) The position of the fish is –5 m.
(f) The position of the stone is –15 m.
Show these expressions using integers:
(a) A pupil got no mark in a Mathematics test _________.
(b) A team scored two points in a game _________
(c) A pupil climbed a tree 10 m tall _________
(d) My coin fell to the bottom of a swimming pool 2 m deep ______
(e) A shopkeeper made a profit of 2 000 Frw _______ .
(f) A shoeseller made a loss of 300 Frw_______ .
Number Line
A number line is a horizontal line drawn with integers marked along its length.
Positive numbers are marked and written on the right hand side of zero and negative numbers are marked and written on the left hand side of the zero mark as shown below:
• Positive numbers are written with a plus (+) sign in front of them. They may also be written without any sign in front, e.g. +5 = 5.
• Negative numbers are written with a minus (–) sign in front of them, e.g. –4.
The Temperature Scale
The temperature is a degree of hotness or coldness of an object.
The temperature of an object is measured using a thermometer.
• The temperature of freezing water is 0°C.
• The temperature of boiling water is 100°C.
• The temperature of a normal person is about 37°C.
The temperature of an object below 0°C is negative. For instance,
temperature in the Antarctica can reach as low as –89.2˚C.
The inverse or opposite of an integer
The opposite or inverse of an integer has the same absolute value but with
an opposite sign.
The inverse of +1 is –1. The inverse of +5 is –5.
The inverse of –2 is +2. The inverse of –6 is +6.
The inverse of –3 is +3. The inverse of +7 is –7 and so on.
The inverse property of an integer
The inverse property of integers states that any number added to its inverse equals to zero.
Draw a number line. Mark all the integers from –10 to +10. (a) Which integer is 4 steps to the right of +4? (b) Which integer is 5 steps to the left of 0? (c) An integer is 6 steps to the left of 1. What is this integer? (d) If you are at zero (0) and you move 9 steps in the negative direction, where will you be on the number line? (e) An insect moves from -6 to +6. How many steps has it moved?
2. Verify the following:
(a) (+1) + (–1) = 0 (b) (–88) + (+88) = 0
(c) (+100) + (–100) = 0 (d) (+6) + (–6) = 0
3. Name the inverse of each of the following integers: (a) +12 (b) +45 (c) –34 (d) –20 (e) –240 (f) –500
Position of Integers on the Number Line
When locating integers on the number line, we need to take great care of the direction in which we are moving. We need to establish a system that we will follow whenever we are locating positions of integers on the number line.
• Our reference point is the zero (0) mark.
• An addition (+) operation means we move towards the right of 0.
• A subtraction (–) operation means we move towards the left of 0.
• We can imagine our face to represent the positive and the back of our heads to represent the negative.
• For positive integers, we move forwards.
• For negative integers, we move backwards.
In this activity, you will need the following materials for your class:
Long string (about 10 m), cotton thread, manila paper of different colours (red, green and yellow), a pair of scissor, two 6 inch nails, hammer, large cardboard box and a black marker pen.
(i) Using a hammer, fix two nails into the opposite walls of the classroom.
(ii) Make loops on the string and tie the string on the nails across the classroom.
(iii) Cut about 10 pieces of red manila papers, 10 pieces of green manila papers and 1 piece of yellow manila paper (each paper should be about 15 cm by 15 cm).
(iv) The teacher will guide you to write integers from –10 to +10 on the 21 pieces of manila papers. The red papers should be used for negative integers, yellow for zero and green for positive integers.
(v) Each pupil picks a cardboard from the box at random, makes a small hole in the paper and ties it with about 30 cm of cotton thread.
(vi) The pupils will stand in groups according to the colour of papers they have picked. There is only one pupil with a yellow paper on which 0 is written.
(vii) The pupil with the yellow paper ties his/her paper to hang exactly in the middle of the string.
(viii) The other pupils also hang their papers on the correct side of the yellow paper and at the correct positions.
(i) Your teacher will take you to a large play area such as a field.
(ii) Using dry sand, ash or any suitable material, draw a number line on the ground and mark integers on it so that each pupil has a point to stand on the number line. The distance between the integers can be about 60 cm.
(iii) Write negative integers on the red papers, 0 on the yellow paper and positive on the green papers.
(iv) Pick the papers at random and tie a loop on the paper so that you are able to wear it around your neck.
(v) Move to the position on the number line which has the same number on your card.
(vi) During this activity, your name is the number that you are wearing on your neck. Listen carefully as the teacher calls your “name” and follow the instructions well.
(vii) Everyone should be able to know how far he/she has moved and in which direction (positive or negative) he/she has moved.
Distance between two integers
1. Two integers on the same side of zero:
The magnitude of –3 is 3 and the magnitude of +3 is also 3.
Magnitude of a number is the distance from 0 to that number.
If two numbers are on the same side of zero, the difference between them is the difference of their magnitudes.
Example 2.2
What is the distance between the following integers on the number line?
(a) + 2 and +10 (b) 0 and +15 (c) –3 and – 6 (d) –7 and –2
Solution
(a) Magnitude of +2 = 2 and magnitude of +10 = 10. so, distance between +2 and +10 = 10 – 2 = 8 steps
(b) In the same way, distance between 0 and +15 = 15 – 0 = 15 steps.
(c) Magnitude of –3 = 3 and magnitude of –6 = 6. So the distance between –3 and –6 = 6 – 3 = 3 steps.
(d) Distance between –7 and –2 = 7 – 2 = 5 steps.
2. Two integers on opposite sides of 0 on the number line:
If two numbers are on opposite sides of 0 on the number line, the distance between them is the sum of the magnitudes of the two numbers. 36
Example 2.3
What is the distance between the following points on the number line?
(a) –5 and +7 (b) –10 and +10
Solution
Distance between – 5 and +7 = 5 steps + 7 steps = 12 steps.
(b) Two numbers on opposite side of zero magnitude of –10 is 10 and the magnitude of +10 is also 10.
So, the distance between –10 and +10 = 10 + 10 = 20 steps.
Without drawing a number line, state the distance between the following points on the number line; (a) 0 and +3 (b) 0 and –3 (c) –2 and +2 (d) –1 and +1
Comparing integers and Ordering integers
• On a number line integers are always in order from the smallest to the biggest.
• If you touch two integers at a time, the one to the left is always smaller and the one to the right is always bigger.
• Any integer is always bigger than the one to its left on the number line.
• Any integer is always smaller than the one to its right on the number line.
Example 2.4
–10 is less than –9 –7 is less than – 4 –2 is less than 0
1 is less than 6 +6 is less than +10 0 is greater than –4
+3 is greater than –10 +1 is greater than –1
Example 2.5
Arrange +1, –3, –10, –2, –4, 0, +5 in order starting with the smallest to the biggest (ascending order).
Solution
We draw the number line and show the positions of the integers.
Order from the smallest to biggest = –10, –4, –3, –2, 0, +1, +5.
1. Show these expressions using integers:
(a) 3 metres below the ground _________ .
(b) 5 steps forward ________ .
(c) 10 steps backward ________ .
2. Name the inverse of each of the following integers:
(a) –5 890 (b) +100 000 (c) –20 000
3. Without drawing a number line, state the distance between the following points on the number line;
(a) +3 and –1 (b) –2 and +10 (c) –10 and +10
(d) –10 and –6
4. Arrange these integers from biggest to smallest (descending order) –3, –7, 0, +4, –13, –31, +13, +31, –301, +310, +301.
5. Arrange these integers from the smallest to biggest (ascending order) –8, –5, 3, 2, 0, +5, 6, 8, 1
Think!!!
A chameleon wants to move from point A to point B, 10 metres apart. It walks in a very funny way such that within 1 minute, it moves 3 metres forward followed by 1 metre backward. How long does the chameleon take to complete its journey to point B?
Key Unit Competence
A learner should be able to classify numbers and appreciate that one number
may belong to various families of numbers.
Attitude and Values
Appreciate the importance of using square roots, being cooperative and
displaying a teamwork spirit.
Natural Numbers
The counting numbers starting from 1 are called natural numbers.
Natural numbers = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...}
Whole Numbers
Natural numbers together with 0 are called whole numbers.
Whole numbers = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...}
Odd Numbers
Odd numbers are natural numbers which are not exactly divisible by 2. When
divided by 2 it always has a remainder of 1.
The following are examples of odd numbers less than 20.
Odd numbers = {1, 3, 5 ,7, 9, 11, 13, 15, 17, 19}
Even Numbers
Even numbers are natural numbers which are exactly divisible by 2. When
divided by 2 it leaves no remainder.
The following are the even numbers found between 0 and 20.
Even numbers = {2, 4, 6, 8, 10, 12, 14, 16, 18}.
[Note: 20 is not included in this set of numbers and yet it is an even number.This is because we were asked to list even numbers between 0 and 20 and not
0 to 20.]
In this class activity, you are going to play
a game which will help you to learn how
to classify numbers as whole, natura odd
and even numbers.
• The teacher will give you a flashcard
with a number written on it.
• Look at your number carefully and
remember it throughout this activity
(game).
• When the teacher calls out the class to which your number belongs, you will run and stand on the line drawn.
List all the odd numbers between 0 and 100.
2. List all the even numbers from 0 up to 100.
Think!!!
I am an odd number. Take away one letter and I become even. What number am I?
Square Numbers
This is the number obtained when one number is multiplied by itself.
Study the multiplication table below and use it to identify all the square numbers less than 60.
So the set of square numbers upto 100 = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}.
Work together in groups and list all the square numbers between 0 and 410.
Square root of a number
Finding the square root of a number is the inverse operation of squaring the number.
The symbol used for the square root of an unknown number is
Evaluate the following by finding the square root.
Complete the multiplication table below and use it to list all the square numbers less than 100.
Prime Numbers
A prime number is any number with only two factors; one factor being 1 and the other one being itself.
Examples of prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, ...
List all the prime numbers between 100 and 150.
Composite Numbers
This is a natural number greater than one which has more than two factors.
In fact a composite number is a natural number greater than 1 which is not a prime number.
The following are examples of composite numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, …
1. What is the smallest prime number?
2. What is the smallest even number which is a prime number?
3. What is the smallest odd number which is prime?
4. List the first 10 composite numbers.
I am an even number. I am a counting number and I am a prime number too.
Who am I and what is my square?
Multiples of a Number
The multiples of a whole number are found by taking the product of any counting number and that whole number.
For example:
So, the multiples of 5 are 5, 10,15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, ….
List the first 10 multiples of the following numbers:
(a) 4 (b) 6 (c) 7 (d) 8 (e) 9 (f) 11 (g) 12
Factors of a Number
The factors of a number are all those whole numbers that can divide evenly to the number and leave no remainder.
The greatest factor of a number is the number itself and the smallest factor of a number is 1.
Work together in groups and list all the numbers between 1 and 30. Which numbers are not multiple of 3
Example 3.1
List all the factors of 18.
Solution
• We already have two factors of 18 namely 1 and 18. 1 is the smallest factor and 18 is the biggest factor.
• Try dividing 18 by numbers 2, 3, 4, 5 ...
• 18 ÷ 2 = 9 and so 2 is the second smallest factor and 9 is the second biggest factor.
• Continue dividing with different numbers until you get all the factors.
• So, the factors of 18 arranged in ascending order are 1, 2, 3, 6, 9, 18.
1 List all the factors of the following numbers:
(a) 8 (b) 12 (c) 24 (d) 36 (e) 48
(f) 64 (g) 45
2. I think of a number. When I multiply it by itself, the answer is 100. What is the number?
Lowest Common Multiple (LCM)
LCM means the Lowest Common Multiple. So, the LCM of two numbers is the smallest multiple which is common to both numbers.
Example 3.2
Find the LCM of the following numbers:
(a) 3 and 4 (b) 5 and 12
(c) 12 and 15 (d) 20 and 65
Solution
Method 1
(a) LCM of 3 and 4
Multiples of 3 = {3, 6, 9, 12, 15, 18, 21, 24, 27, .…}
Multiples of 4 = {4, 8, 12, 16, 20, 24, 28…}
The common multiples are 12, 24, ……..
The smallest common multiple is 12 and so the LCM of 3 and 4 = 12.(b) LCM of 5 and 12
Multiples of 5 = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, ...}
Multiples of 12 = {12, 24, 36, 48, 60, 72, 84, 96, 108, ...}
The lowest common multiple is 60.
So, the LCM of 5 and 12 = 60.(c) LCM of 12 and 15
Multiples of 12 = {12, 24, 36, 48, 60, 72, 84, 96, 108, ...}
Multiples of 15 = {15, 30, 45, 60, 75, 90, 105, 120, 135, ...}
The lowest common multiple is 60.
So, the LCM of 12 and 15 = 60(d) LCM of 20 and 65
Multiples of 20 = {20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220,
240, 260, ...}
Multiples of 65 = {65, 130, 195, 260, 325, 390, ...}
The lowest common multiple is 260.
So, the LCM of 20 and 65 = 260.Method 2
Another way of finding the LCM of two or more numbers is to first express
them in terms of prime factors. Let’s use this method to find the LCM of 12
and 15 and see whether we shall get 60.Solution
Factorize 12 and 15 in terms of prime factors.
The LCM can be got by multiplying all the prime factors of 12 and 15 in the
first column on the left.
So, LCM of 12 and 15 = 2 × 2 × 3 × 5 = 60.
1. Evaluate the following by finding its square root:
2. Find the first five multiple of the following numbers: 3, 6, 5, 9, 10
3. List the first 10 prime numbers.
(a) Circle all the square numbers.
(b) Using a pencil, tick all the composite numbers.
4. Find the HCF and LCM of the following numbers:
(a) 2 and 4 (b) 4 and 5 (c) 3 and 6 (d) 4 and 10
(e) 12 and 14 (f) 23 and 46
For more online support visit
https://www.superteacherworksheets.com/least-common-multiple.htm
Key Unit Competence
By the end of this unit, a learner should be able to explain the meaning of fractions, add and subtract same-denominator fractions, multiply and divide fractions accurately.
Attitude and Values
Appreciate the importance of accuracy in carrying out operations on fractions and develop the spirit of sharing with others.
Fraction
A fraction is a part of a whole number. A fraction is made up of 2 numbers. The top number is called the numerator and the bottom number is called the denominator. For example: In the fraction 3/4, 3 is the numerator and 4 is the denominator.
How Fractions Arise?
Fractions are obtained when a whole number is divided into two or more than two parts. The figures below show an orange divided into 8 equal parts.
If one part is removed and eaten, then this is represented by the fraction 1/8.
The part which has not yet been eaten is represented by the fraction 7/8.
The figure aside shows a flower. The flower has four equal parts. 2 parts are coloured red, 1 yellow and 1 part is not coloured, i.e., it is white. This can be represented using fractions:
The red part is represented by the fraction 2/4, the yellow part is represented by the fraction 1/4 and the white part is represented by the fraction 1/4.
Mixed Fractions
In this class activity, you will identify the numerator and denominator in the following fractions.
You will work in groups of 4.
• This is a clock face with so many numbers, some of which are fractions.
• Write down all the fractions shown on the clock face above.
• For each fraction you have written above, write down the numerator and denominator.
• Compare your results with other classmates in class.
• Present your answers to the teacher for marking.
In this class activity, you will write fractions when you are given the numerator and denominator.
You will complete this activity in groups of 4. Complete the table in your note book, writing down the fraction.
A farmer has a bag of popcorn, a hen and a fox. He wants to cross a river in a boat. The boat can only take him and only one of the three items he has. The problem is that the hen can eat popcorn and the fox can eat the hen. How does he cross the river without anything getting eaten up?
Look at the following figures. Use coloured pencils or crayons to colour or paint as instructed below:
Reading a Fraction
Look at the following fraction: 2/5
This fraction can be read as:
- Two-fifth or
- Two over five or
- Two divided by five.
1/2 is read as “half” or “one over two” or “one divided by two”.4/5 is read as “four-fifth” or “four over five” or “four divided by five”.
1/10 is read as “one-tenth” or “one over ten” or “one divided by ten”
7/1000 is read as “seven-thousandth” or “seven over one thousand” or “seven divided by one thousand”
(a) In this class activity, you will work in groups of 6 learners.(b) Three of them will write a half, a fifth and a third on their respective cards as shown below.
(d) Now, make pairs of similar values of cards.(e) Ask your teacher to check the results.
Shade the parts represented by the given fraction.
Fill in “numerator” or “denominator” to complete the following.
Comparing fractions with the same denominator
When comparing fractions of the same denominator, we look at the numerators. The fraction with the bigger numerator is the greater fraction.
For example10/17 is greater than 8/17.
In this class activity, you will work in groups of 5 learners.
(a) Cut the orange given to you into 4 equal parts. Each part shall be called a slice.
(b) Display the slices on a table and count them to be sure that they are 4.
(c) What fraction represents all the slices you have counted?
(d) Put the slices in 3 groups labelled A, B and C. In A put 1 slice, 2 slices in B and 1 slice in C.
(e) What fraction represents the number of slices in each group? Answer this question by completing the table below:
(f) Which group has the greatest number of slices? (g) Which group has the smallest number of slices?
In this activity, you will compare the given fractions using the correct comparison symbols; < (less than), > (greater than) or = (equal to).
1. Fill in ‘is greater than’ or ‘is less than’ to complete the following statements.
2. Shade the fraction given and use the symbols >, < or =.
Addition of Fractions with the Same Denominator
To add fractions of the same denominator add only the numerators and use the denominator once.
(a) Draw a circle of radius 10 cm on a manila paper.
(b) Mark the centre of the circle using letter O.
(c) Draw two lines through the centre of the circle to divide the circle into four equal parts. Each of these parts is called a sector.
(d) Label the three boxes A, B and C.
(e) Cut out the outline of the circle along the lines drawn.
(f) What fraction does each sector represent? Your answer should be
(g) In box A put one sector, put two sectors in B and one sector in C.
(i) Pick the sectors in box A and box B. Add them together. What total fraction do you get?
(j) Add the sectors in A and C together. What fraction do you get?
Mr. Kamanzi came back home from Kigali with 24 apples. He decided to share all the apples between his two sons Ted and Ronald. Ted was given a quarter of a half of the apples and Ronald was given a half of a quarter of the apples. Which son got more apples?
Add the following fractions:
Subtraction of Fractions with the Same Denominator
When the denominator is common. Subtract only numerators and take the common denominator.
Note: Remember always to reduce the answer to its simplest form.
Subtract the following fractions:
Real Life Problems involving Fractions
Multiplication of Fractions by Whole Numbers
Get 8 bottle tops
Arrange them into four equal groups.
What is a quarter of eight? Use your arrangement to get the solution.
This is how we get the answer.
The box below is divided into four equal parts. The shaded part is 4/1
Multiply the following fractions by the given whole numbers:
Multiplication of Fractions by Fractions
Two fractions can be multiplied together using a fraction grid.
(a) Draw a rectangle as shown below.
(b) Separate the rectangle into three equal parts using two vertical lines.
(c) Shade two of these parts. The shaded area (blue) represents the fraction 2/3.
(d) Now divide the rectangle into four equal parts using three horizontal lines.
(e) Shade three of these parts to represent 3/4.
(f) Now count the total number of parts in the rectangle. You will find 12 parts.
(g) Count the number of parts that have been double shaded. There are 6 out of 12 parts which are double shaded. (This is represented by the red parts)
Draw a rectangle as shown below.
Divide the rectangle into parts, shade to show the fractions 2/3 and 2/5. Count the number of parts that have been double shaded. We see that 4 out of 15 parts have been shaded.
So since 4 parts are shaded twice, then the answer is 4/15.
Let us multiply the fractions below and reduce them to their simplest form:
Division of Fractions by Whole Numbers
To divide a fraction by a whole number:
Step 1: Multiply the denominator by the whole number.
Step 2: Simplify the fraction where necessary.Example 4.18
1/2 ÷ 3Solution
Step 1: Multiply the bottom number of the fraction by the whole number:
Step 2: Fraction is already as simple as possible, so no need for step 2.
Compare 1/2 and 1/6. Which one is smaller? 1/3 is smaller than 1/2.
Work out the following questions:
Real Life Problems involving Fraction
How many half litre bottles can be filled from a twenty litres jerrycan?
- In the following diagrams, white parts are not shaded. The coloured parts are shaded. Count the shaded parts carefully and write down the fraction shown by the shaded part (s).
- Given 21/30, 30 is the ________ and 21 is the ________ .
- Given 59/60, 60 is the ________ and 59 is the ________ .
- Arrange the fractions provided below in descending order (starting with the biggest to the smallest).
- Add the following fractions:
- Subtract the following fractions:
- A day is made up of 24 hours. Jane spends 9/24 of the day studying and 11/24 of the day sleeping. What fraction of the day does he spend sleeping and studying altogether?
- Manzi ate 1/5 of a cake at breakfast and 3/5 at lunch. What part of the cake did he eat?
- Multiply the following fractions by the given whole numbers:
- What is 2/5 of 4/9?
- Find the product of 13/20 and 2/3
- Find the product of 13/20 and 4/5
- Find 8/9 of 3/4
- Work out: 4/7 of 14
- Multiply 9/11 by 1/3
- Work out the following questions:
- A mother has a cake. She decides to share it equally among her 5 children. What fraction does each child get?
- Half of a pawpaw is shared equally by 2 people. What fraction of the pawpaw does each get?
- In a class there are 30 students. Two-thirds of these are girls. How many girls are in the class?
- A soccer stadium has 20 000 spectators. Half of the spectators are wearing red jerseys. How many spectators are wearing red jerseys?
- Divide 9/2 by 9.
- My dad earns Frw 250 000 from the farm every month. He spends 1/5 of this on food, 3/5 on school fees and saves the rest.
(a) How much money does he spend on food?
(b) What fraction of his salary is saved?
(c) How much does he save per month?
23. Ineza gave 83 of her birthday cake to her brother and she kept 81. Workout the part, how much more is given to his brother?
24. A plot of land in a certain town costs Frw 100 000. Peter bought one-tenth of the plot. What amount of money did he pay for the plot?
25. In our school, a lesson lasts 40 minutes. We spend 81 of the lesson revising the previous exercise. How long do we spend on revision per lesson?
26. A milk container can hold 15 litres of milk. The milk container falls so that one third of the milk pours out. How many litres of milk remains in the can?
Key Unit Competence
A learner should be able to add, subtract and compare decimal numbers using place values of decimals up to 2 decimal places.
Attitudes and values
Develop personal confidence in the use of decimal numbers and appreciate the importance of decimal fractions in comparing and sharing.
Decimal number
A decimal number is any number which contains a decimal point.
A decimal number is a number which has a decimal part separated from the integer part using a decimal separator called the decimal point.For example, 28.53 is a decimal number.
Activity 5.1
i. Draw a horizontal line of about 12 cm on your notebook using a ruler
and sharp pencil.
ii. Show the integers –4, –3, –2, –1, 0, +1, +2, +3 and +4 on the number line.Activity 5.2
In this class activity, you will find some numbers:
(a) Divide the following numbers by 10:
- 1400
- 140
- 14
- 1.4
(b) What number is exactly:- between 0 and 100?
- between 0 and 10?
- between 0 and 1?
Decimals on a number line
Consider the number line below. The number line shows negative and positivenumbers from –5 to +5.
However, if we zoom into the space between 0 and 1, then we create new numbers which are not whole. The space between 0 and 1 is divided into 10 equal parts. The decimal numbers 0.1, 0.2, 0.3, 0.4, ...... , 0.9 are produced. Decimals like 0.25 and 0.75 can easily be located as shown below.
By further zooming into the space between 0.1 and 0.2, we can form new decimal numbers as shown below.
Activity 5.3
In this class activity, you will divide the space between 2.0 and 3.0 into ten equal parts.
i. Draw a horizontal line of about 10 cm on your notebook using a ruler and sharp pencil.
ii. Mark the start of the line with the decimal number 2.0.
iii. Mark the end of the line with the decimal number 3.0.
iv. Mark the line drawn into 10 equal parts, each 1 cm long.
v. Show the decimal numbers 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8 and 2.9 on the number line.
Activity 5.4
In this class activity, you will locate the positions of the given decimal numbers: 8.11, 8.16 and 8.19.
i. Draw a horizontal line in your notebook using a sharp pencil. Length of the line should be 10 cm.
ii. Mark the line at intervals of 1 cm so that the line has 10 equal parts.
iii. Write the decimal number 8.10 on the first mark and the decimal number 8.20 on the last mark.
iv. Carefully study the number line you have drawn and write the correct decimal numbers on the marks drawn.
v. Show the positions of the decimal numbers 8.11, 8.16 and 8.19.
vi. Present your work to the teacher for evaluation.
Exercise 5.1
Show the following on the number line:
(a) 0.6, 0.8, 0.3, 0.1
(b) 0.43, 0.48, 0.25, 0.15
(c) 0.3, 0.1, 0.5, 0.9, 0.2
(d) 0.51, 0.50, 0.48, 0.55
Place Value
Consider the decimal number 78.46
78.46 is read as “seventy eight point four six”.
It is wrong to say “seventy eight point forty six”.
We can see that the place value for 7 is ‘tens’, 8 is ‘ones’ or ‘units’, 4 is ‘tenths’,and 6 is ‘hundredths’.
Value of 7 is 7 × 10 = 70
Value of 8 is 8 × 1 = 8
Value of 4 is 4× 1/10 = 0.4
Value of 6 is 6 ×1/100= 0.06Activity 5.5
i. Let two pupils A and B stand to the right of a tree.
ii. Also, two pupils C and D should stand to the left of the tree.
iii. If we now consider the tree as a decimal point, let each pupil mention his/her place.Example 5.1
Write the places of the underlined digits in the decimal numbers given
below:(a) 0.24 (b) 5.06 (c) 13.56
Solution(a) Tenths (b) Ones (c) Tens
Exercise 5.2
1. Read the following decimal numbers and write them in words.
(a) 87.09 (b) 12.2 (c) 0.03 (d) 23.56
(e) 19.19 (f) 0.10 (g) 45.452. Write the following decimal numbers in figures:
(a) One hundred forty-four point seven.
(b) Twelve point one two.
(c) Seven point seven.
(d) Eighty point seven one.Comparing Decimals
When comparing decimal numbers, we may use =, < or >.
Example 5.2
Compare the decimal numbers 9.25 and 9.7.
Solution
1. Align the decimal points.
2. Fill in the empty place with zero.
3. Compare the digits from left to right until they are different.
Example 5.3
Compare the two numbers: 452.78 and 452.21. Which one is greater?
Solution
Arrange the numbers in a table like this:
The first three digits are the same in both decimal numbers. But in the tenths’ column, 7 is greater than 2. So, 452.78 > 452.21.
Example 5.4
Arrange the following numbers in ascending order:
0.46, 0.64, 0.9, 0.09, 0.57, 0.75
Solution
• All these numbers begin with zero.
• Now we compare the second numbers.
• We can see that the smallest number is 0.09, followed by 0.46......
• 0.57 is smaller than 0.64.
• 0.75 is smaller than 0.9.Therefore, Ascending order: 0.09, 0.46, 0.57, 0.64, 0.75, 0.9.
This can also be written as 0.09 < 0.46 < 0.57 < 0.64 < 0.75 < 0.9.Example 5.5
Arrange the following numbers in descending order:
8.6, 7.66, 7.6, 0.76, 0.67, 0.86, 6.08.Solution
Descending order: 8.6, 7.66, 7.6, 6.08, 0.86, 0.76, 0.67.
Alternatively, 8.6 > 7.66 > 7.6 > 6.08 > 0.86 > 0.76 > 0.67.Exercise 5.3
1. Arrange the following decimal numbers from the smallest to the largest (Ascending order)
(a) 3.5, 3.79, 3.42, 3.57, 3.7, 3.62.
(b) 5.7, 5.64, 5.8, 5.4, 5.79, 5.72.
(c) 5.0, 4.7, 4.8, 4.9, 4.3, 4.75
(d) 1.02, 1.7, 1.12, 1.66, 1.71, 1.1
(e) 3.1, 2.5, 2.49, 2.8, 3.48, 2.522. Arrange the following decimal numbers from the largest to the smallest (Descending order).
(a) 8.1, 7.9, 7.92, 8.43, 7.89, 7.97
(b) 6.8, 7.23, 7.32, 6.59, 6.92, 7.02
(c) 2.01, 2.10, 2.63, 2.36, 1.4, 1.7
(d) 3.8, 2.77, 2.75, 3.34, 2.9, 3.4
(e) 6.4, 7.4, 4.7, 4.6, 4.06, 7.041. Arrange the following decimal numbers from the smallest to the largest (Ascending order)
(a) 3.5, 3.79, 3.42, 3.57, 3.7, 3.62.
(b) 5.7, 5.64, 5.8, 5.4, 5.79, 5.72.
(c) 5.0, 4.7, 4.8, 4.9, 4.3, 4.75
(d) 1.02, 1.7, 1.12, 1.66, 1.71, 1.1
(e) 3.1, 2.5, 2.49, 2.8, 3.48, 2.522. Arrange the following decimal numbers from the largest to the smallest (Descending order).
(a) 8.1, 7.9, 7.92, 8.43, 7.89, 7.97
(b) 6.8, 7.23, 7.32, 6.59, 6.92, 7.02
(c) 2.01, 2.10, 2.63, 2.36, 1.4, 1.7
(d) 3.8, 2.77, 2.75, 3.34, 2.9, 3.4
(e) 6.4, 7.4, 4.7, 4.6, 4.06, 7.04Addition and Subtraction of Decimal Numbers
Adding decimal numbers works exactly the same way as adding whole numbers.
You just line up the decimal points as shown below. For example,
Add
Subtraction works well in the same way as subtracting whole numbers.
For example, subtract 8.97 – 2.82
Activity 5.6
In this class activity, you will find any two decimal numbers which add up to the given whole number:
(a) Find any 3 pairs of decimal numbers which add up to 10.
{e.g. 8.5 + 1.5, 5.5 + 4.5, and 3.55 + 6.45}(b) Find any 3 pairs of decimal numbers which add up to 1.
{e.g. 0.25 + 0.75, 0.8 + 0.2 and 0.77 + 0.23}Example 5.6
Add 528 + 7.49
Solution
We can write 528 as a decimal number by putting a decimal point after 8
and adding zeroes. So, 528 can be written as 528.00
Example 5.7
Add the following decimal numbers:
(a) 206.1 + 223.9 (b) 404.1 + 247.4 (c) 665.2 + 567
(d) 435.7 + 65.1 (e) 299.72 + 0.08
Example 5.8
Subtract: 3.8 – 1.26
Exercise 5.4
Work out the following:
(a) 4.7 + 4.7 (b) 2.78 + 3.62 (c) 20.9 + 25.7
(d) 20.23 + 40 (e) 25.26 + 0.72 (f) 400.2 + 400
(g) 18.6 – 8.6
(j) 80.02 – 0.02 (k) 19.82 – 5.28 (l) 20.82 – 6.27
(m) 20.9 + 25.7
Real life problems involving decimals
Example 5.9
Ogolla ate 0.5 of his apple in the morning and 0.2 of it in the afternoon. How much of his apple did he ate altogether?
Therefore, he ate 0.7 of his apple.
Example 5.10
1. Opio bought 4 metres of cloth and used 2.3 metres for making shirts.
Find the length of the remaining cloth.
2. Nankinga weighs 94.2 kg and his brother Hamidu, 87.5 kg.
What is the difference between their weights?
- Write the following decimal numbers in figures:
(a) Three hundred ninety-nine point nine nine.(b) One thousand seventy point seven zero.2. Compare the following decimal numbers by writing the symbols =, < or > in the box.
3. Work out the following:(a) 84.0 + 79.3 (b) 9.1 – 6.1 (c) 53.3 – 23.2
(d) 92.7 – 20.7 (e) 8.9 – 0.9 (f) 689.6 – 609.8
(g) 100.5 – 50.5
4. A boy walks 2.5 km to school and then 1.5 km to his friend’s home.What is the total distance covered?
5. The masses of four children in our class are 34 kg, 43.4 kg, 36.3 kg and 38.2 kg. What is their total mass in kg?
6. A lesson starts at 8:30 a.m. and lasts for 50 minutes. At what time does it end?
7. The lengths of four rivers are 2.5 km, 4 km, 10.45 km and 0.95 km respectively. What is the total length of the four rivers? By how much is the longest river more than the shortest river?
8. In an athletics school race competition, the time taken to complete 100 m race is given below:
(a) Who won the race? Explain why.(b) Who came last in the race? Explain why.
(c) For how long did the first competitor wait for the last person to finish the race?
9. If x = 23.23 and y = 32.32, find the value of:(a) x + x
(b) x + y
(c) y – x.
10. By how much is 18.5 greater than 14.6?
11. I am 9 years old. My elder sister is 14.5 years old. My mother is 31.5years older than me.
(a) How old is my mother?
(b) What is the total of all our ages?
12. My wrist watch displays time using figures. The time right now is shown as 8:15 a.m. What time will my watch show after 15 minutes from now?- 1400
Key Unit Competence
A learner should be able to know the meaning of length, conversion between units of length and apply them in solving mathematical problems related to daily life situations.
Attitudes and values
Learners should appreciate the importance of metric measures in daily life and recognise the importance of using, measuring tools correctly.
Meaning of Length
Distance between two points is known as length.
Instruments for Measuring Length
There are many instruments used for measuring length. These type of instruments are used depends on the size of the length to be measured. Some instruments used for measuring length/distances are shown below;
1. Ruler: It is used to measure short distances like length of a line in a book, length of a table, classroom, height of a door.
2. Tape measure: It is used for measuring longer distances like length of a field, height of a person, length of a cloth, etc.
3. Yard stick: It is a flat wooden board with markings at regular intervals. It is used for measuring small lengths.
4. Micrometer screw gauge: It is used for measuring very small distances such as diameter of a pin, thickness of a paper. A micrometer screw gauge appears like shown below:
Other instruments used for measuring length are vernier calliper, Architect’s scale, etc.
We can measure the length using pace, handspan, foot span, cubit, etc. These units vary from persons to persons as they have different pace, handspans,etc. This method is used to measure small distances.
For example, the length of a table is 6 hand spans, the length of a carpet is 8 foot spans.
• Stand straight up on the floor.
• Place the metre rule near your foot so that it is parallel to the side of your body.
• Mark a point on your body where the end of the metre rule touches.
• Now you have your own metre rule and be sure to tell the length of 1 metre.
• Cut the stick provided so that its length is 1m.
• How many times does your foot fit into a metre?
• You can now use your foot to measure the length of a classroom.
Check and write the lengths/heights of the following objects present in your home.
2. Check and write the length/height of the following.
Standard Units of Measuring Length
The standard unit for measuring
length is metre. It is denoted by ‘m’.
The other units of length are the kilometre (km), hectometre (hm), decametre (dam), decimetre (dm), centimetre (cm) and millimetre (mm).Metric-unit Prefixes
Metric prefixes are very useful in converting units of quantities. The main metric prefixes dealt with at this level are the kilo, hecto, deca, deci, centi and milli.
Important mnemonic:
From the above table we can see that;
• One kilometre = 1 km = 1000 m.
• One hectometre = 1 hm = 100 m
• One decametre = 1 dam = 10 m
• One decimetre = 1 dm = 0.1 m ( a tenth of a metre)
• One centimetre = 1 cm = 0.01 m = (hundredth part of a metre)
• One millimetre = 1 mm = 0.001 m = (a thousandth of a metre)
Exercise 6.2
Estimate the length of the following in metres.
(a) length of a car. (b) length of a bus.
(c) length/height of your handspan. (d) length of your pace.Estimate the length of the following in metres.
(a) length of a car. (b) length of a bus.
(c) length/height of your handspan. (d) length of your pace.Estimation of Length
In this section, we will study to find the length of objects by observing them.
For example, the length of a notebook is about 12 cm, the length of a pencil is about 5 cm, etc.
Estimate the length of the following in metres.
(a) length of a car. (b) length of a bus.
(c) length/height of your handspan. (d) length of your pace.
Exercise 6.3
Since we have practically seen the length of a mm, cm, dm, m, dam, hm and km, it is now time to see how well you can estimate length.
Estimate the following lengths:
(a) Length of your foot (in cm).
(b) Length of your classroom (in m).
(c) Length of your pencil in mm (or cm).
(d) Width of your exercise book (in cm).
(e) Length of your table (in dm).
(f) Height of your best friend in class (in cm or m).Conversion of Units
When converting from one unit of length to another, we can easily use a conversion table.Conversion table for length
From the conversion table, you can see that;
• 1 km = 10 hm, 1 km = 100 dam, 1 km = 1000 m
• 1 m = 10 dm, 1 m = 100 cm, 1 m = 1000 mm,
• 1 hm = 0.1 km, 1 dam = 0.01 km, 1 m = 0.001 km
• 1 dm = 0.1 m, 1 cm = 0.01 m, 1 mm = 0.001 m.Example 6.1
Convert the following units of length:
(a) 2 km to m (b) 2 km to hm (c) 10 cm to mm
(d) 3 dam to m (e) 40 m to mm (f) 50 dm to cm
Solution
(a) Use the conversion table (b) Use the conversion table
So, 2 km = 2 x 1 000 m = 2 000 m So, 2 km = 2 x 10 hm = 20 hm
Example 6.2
Exercise 6.4
1. Complete the following:
(a) 40 km = ............... m (b) 350 cm = ............. mm
(c) 3 hm = .................. m (d) 40 cm = .............. dm
(e) 1 m = ................. mm (f) 2 900 mm = ......... cm
(g) 3 000 000 mm = ............... m2. Use the conversion table to convert the following units of length as instructed:
(a) 2 m = ................ cm (b) 3 km = ............... dam
(c) 2.5 m = ............. cm (d) 46 cm = .............. mm
(e) 2 000 m = ............... km (f) 50 mm = ............... cm
(g) 4500 dm = .............. dam.Addition and Subtraction of Length
The process of addition or subtraction of length is exactly similar to the addition or subtraction of ordinary numbers.
Example 6.3
Add the following:
(a) 3 km + 200 m = ................... m
(b) 50 hm + 40 dm = ................... dm
(c) 16 m + 14 dm = ................... m
(d) 76 hm + 4 km = ................... mSolution
(a) Use the conversion table
Example 6.4
Subtract:
1. 226 cm – 105 cm
2. 3 m 15 cm – 1 m 35 cm
3. 125.2 mm – 87.6 mm
Solution1. 226 cm – 105 cm = 121 cm
2. 3 m 15 cm = 3 × 100 cm + 15 cm
= 300 cm + 15 cm
= 315 cm
1 m 35 cm = 100 cm + 35 cm
= 135 cm
Therefore, 315 cm – 135 cm = 180 cm3. Follow steps as we follow in decimal subtraction.
Exercise 6.5
Workout the following:
(a) 30 km + 4 000 m = ............... m
(b) 3 600 m + 44 hm = ............... km
(c) 65 hm – 25 dam = ............... m
(d) 30 km – 4 000 m = ............... hm
(e) 40 hm – 200 dam = ............... dm
(f) 500 m + 28 000 dm = ............... cmThe Perimeter
The perimeter of a closed figure is the distance around the figure. Since the standard unit of distance or length is the metre (m), then the unit of perimeter is the metre. However, perimeter can also be expressed in km, hm, dam, dm, cm or mm.
Example 6.5
The diagram below shows a piece of paper. Length AB = 6 cm, length BC = 2 cm, length CD = 6 cm and length DA = 2 cm. Calculate the distance all round the piece of paper.
Solution
Perimeter = length of (AB + BC + CD + DA)
= 6 cm + 2 cm + 6 cm + 2 cm
= 16 cm.Example 6.6
The diagram below shows an isosceles piece of cake PQR. The lengths of the sides of the cakes are shown on the diagram. What is the distance all round the figure PQR?
Solution
Perimeter = Distance around PQR = PQ + QR + RP = (3 cm + 4 cm + 4 cm)
= 11 cm.Example 6.7
The diagram below shows a birthday card for my younger sister Cathy. The length of each side of the card is 5cm. What is the total distance all round the card?
Solution
Perimeter = Length of (PQ + QR + PR)
= 5 cm + 5 cm + 5 cm
= 15 cmExample 6.8
What is the distance all around this plot of land? Express your final answer in metres.
Solution
The sides are given in different units. We need to convert all units to metres using the conversion table.
From the conversion table
2 000 mm = 2 m
20 dam = 200 m
300 cm = 3 m
10 dam = 100 m
20 000 mm = 20 m
So the required distance = 2 m + 200 m + 3 m + 100 m + 20 m + 90 m =415 m
Exercise 6.61. Calculate the perimeter of the figure shown below.
2. The figure below shows a composite shape. It is formed by joining rectangular shapes 1, 2 and 3.
Application of Length
Sometimes, you find situations where you need to apply length in order to solve them. Study the following examples.
Example 6.9
Trees are planted at an intervals of 10 m. If ten trees are planted, find the distance from the first to the last tree.
SolutionFirst find the number of spaces
Number of spaces = Total number of trees – one tree = 10 – 1
= 9 spaces
1 space = 10 m
9 spaces = (10 × 9) m = 90 mThe distance is 90 metres.
Example 6.10
A rectangular garden 50 m by 30 m is to be fenced with poles placed at an interval of 5 m. Find the amount of money required to fence the garden at 3 000 Frw per pole.
Solution
Number of poles;
Along the length = 50/5 =10 poles
Along the width = 30/5 = 6 poles
Total number of poles = (10 + 6) × 2 = 16 × 2 = 32 poles
Cost of 1 pole = 3 000 Frw
Cost of 32 poles = 32 × 3 000 Frw
= 96 000 FrwNote: Number of poles around a circular garden = Circumference/Interval
Activity 6.5
Eleven bundles of wires, each of 125 cm long were needed from Kabuye’s house to the electric pole. How far was Kabuye’s house from the electric pole?
Assessment Exercise
1. Estimate the following lengths:
(a) Height of your teacher (in m).
(b) Length of your football field at school (in m).
(c) Height of the tallest tree at your school (in m).
(d) Length of your handspan (in cm).2. Workout the following:
(a) 1 100 mm + 1 100 cm = ............... cm
(b) 800 hm + 5 000 dam = ............... km
(c) 1 km + 1 hm + 1 dam + 1 m + 1 dm + 1 cm + 1 mm = ................... mm3. Find the perimeter of the shape shown in the diagram below:
4. Electric poles are fixed at intervals of 50 m apart. If 101 poles are fixed, find the distance from the first pole to the last pole.
5. How many poles are required to make a circular fence of 45 m if the poles are 5 m apart?
6. Othieno has to fence his rectangular garden of 45 m by 30 m.
(a) Find the number of poles he will require if the poles are 5 m apart?
(b) If each pole costs 4 500 Frw, how much does he require to buy the poles?7. Find the number of trees required to plant around a circular pond of length 28 m at intervals of 4 m apart.
8. Luzige’s land is in a shape of trapezium as shown:
He intends to fence it with poles 5 m apart. How many poles does he require?
9. Workout:
(a) 1.25 cm + 7.75 cm
(b) 9.75 dm + 2.45 dm
Key Unit Competence
A learner should be able to understand capacity, convert between units of capacity and apply them in solving mathematical problems related to daily life situations.
Attitudes and values
• Show an ability to properly use a range of materials to measure different liquids in daily life.
• Being honest and trustworthy when measuring different capacities.
• Being able to show respect for one another when working in groups.Understanding Capacity
In our daily conversation, we usually make statements like:
•This is a small plane and that is a big plane.
• My cup is small but Dad’s cup is big.
• An orange is smaller than a pineapple.
• A lion is smaller than an elephant.
All the above statements compares the volume or capacity of one object with another.
Volume is the space occupied by a given object.
Capacity, on the other hand, is the amount of liquid or solid an object can hold.
Collect empty containers and compare their capacities by filling water or any other liquid.
Which has more capacity/volume?
Units of volume
Consider a container in the form of a cube as shown in the diagram below.
Each side of the container is of length 1 cm.
Capacity of the container = length × width × height = 1 cm × 1 cm × 1 cm = 1 cm3
(a) Measure the length (l ), width (w), and height
provided. Record your results in millimeters (mm).
(b) Calculate the volume of one matchbox using the formula;
V = l × w × h.
(c) Pile the matchboxes so as to form a big cuboid of matchboxes.
(d) What is the volume of the big box you have formed?
(e) Can you design other different big boxes using the same matchboxes?
1. The figure below shows several small cubes of different colours joined together. Each small cube has a volume of 1 cm3.
(a) How many small cubes are there altogether?
(b) What is the total volume of all the cubes?
2. Find the volume of the solids if each cube has a volume of 1 m3.
Estimating capacities of different containers
In this class activity, you are going to discuss the statements given withyour classmates.
State whether you agree or not. You may experiment (if possible) in your classroom.
(a) The capacity of a teaspoon is about 5 ml or 5 cm3. The capacity of an ordinary glass for drinking water is about 300 ml or 300 cm3 or 0.3 liter.
(b) The capacity of a mug is about 0.5 liter. The capacity of a water jug is
about 2 liters.
(c) The capacity of a small jerrycan is about 5 liters. The capacity of a big
jerrycan is about 20 liters.
(d) A large water tank has a capacity of about 1 000 litres.
Estimate the capacities of the following containers:
What is full of holes and yet it holds lots of water?
Conversion of units
Conversion table for capacity
We can see that,
1kilolitre = 1 kl = 1 000 l
One hectolitre = 1 hl = 100 l
One decalitre = 1 dal = 10 l
One litre = 1 l = 10 dl
One decilitre = 1 dl = 0.1 l (a tenth of a litre)
One centilitre = 1 cl = 0.01 l (a hundredth of a litre), e.t.cExample 7.1
Change the following units of capacity as instructed below:
(a) 20 l to ml (b) 200 l to hl (c) 20 kl to l
(d) 50 kl to dal (e) 100 dl to ml (f) 35 hl to dal
(g) 7 800 cl to lSolution
We construct a conversion table to enable us convert these units.
(a) 20 l = 20 000 ml (b) 200 l = 2 hl (c) 20 kl = 20 000 l
(d) 50 kl = 5 000 dal (e) 100 dl = 10 000 ml (f) 35 hl = 350 dal
(g) 7 800 cl = 78 l
A cow produced milk as follows;
Monday - 2 litres
Tuesday - 5 litres
Wednesday - 3 litres
How much milk did it produce in three days? Write answer in ml.
Express the following in millilitres:
Addition and Subtraction of Capacity
In groups, do the following:
• Measure the capacity of water in a jug.
• Change this into litres.
• Now, measure the capacity of a bucket by pouring water using 1 litre
bottles.
• Compare the capacities. Which has more water? Obtain the total volume
of water in litre.Example 7.2
(a) Add: 900 decalitres + 400 litres
(b) Subtract: 24 litres – 2 400 centilitersSolution
900 dal + 400 l = 9 000 l + 400 l = 9 400 l
24 l – 2 400 cl = 2 400 cl – 2 400 cl = 0.
Example 7.3
(a) 7.50 l + 3.50 l
(b) 6.30 l – 2.30 lSolution
Application of Capacity
Example 7.4
1. A bottle of mineral water has a capacity of 300 ml. How many bottles of
mineral water do I need to make 3 litres?
2. Mutoni sells 11 cans of milk to a milk depot every week. If each can
of milk has a capacity of 20 litres, how much milk does she sell per
month?Solution
1. 3 l = 1 000 × 3 = 3 000 ml. Since 3 l = 3 000 ml, therefore, the number
of mineral water bottles is given by:
3000 ml ÷ 300 ml = 10
2. Number of cans sold per week = 11
Number of cans sold per month = 11 × 4 = 44 cans (1 month = 4 weeks)
Quantity of milk sold in 1 month = 44 × 20 litres = 880 litres.
Study the following figure and answer the given questions:
1. Find how much water should be added so as to fill up the tank fully.
2. How much water was there earlier in the tank. (Dimensions shown in
figure are in centimetres)
1. Add: (a) 3 l 500 ml + 5 l 700 ml (b) 3.25 l + 6.75 l
2. Subtract: (a) 5 l 400 ml – 3 l 250 ml (b) 10.5 l – 2.75 l
3. A restaurant sells 200 l of milk every day. How much milk is sold in 3 months? (assume 1 month = 30 days).
4. A school water tank contains 12 000 l of water. Pupils in the school used 40 hectolitres of water. What amount of water is remaining?
5. The capacity of a milk jug is 750 ml. By how much is this less than a litre?
6. The cost of milk is 150 Frw per mug (cup). Two such cups make up one litre of milk. How much will a person spend on 4 litres of milk?
7. On average, each pupil takes 1.5 litres of milk in our school. How much milk is taken by 100 pupils?
8. A Friesian cow in a certain farm produces 15 litres of milk in the morning and 10 litres of milk in the evening. How much milk does it produce in a week?
9. The price of petrol is 900 Frw per litre. A school bus requires 100 litres of petrol for a school trip. How much is spent on petrol?
For more online support visit www.math-play.com/capacity
Key Unit Competence
By the end of this unit, a learner should be able to convert between units of mass and apply them in solving mathematical problems related to daily life situations.
Attitudes and values
Appreciate the importance of mass measurement in daily life, show respect for one another, and appreciate difference in opinion while working with other people and show fairness while measuring mass.
Estimating Mass
In this section, you will study to estimate the mass of an object by observing it only.
• You will be provided with a stone of mass 500 g (0.5 kg) and another stone of mass 1 kg.Other alternative masses may be provided.
• Feel the mass of 500 g by holding the stone in your hand.
• Feel the mass of 1 kg by holding the stone in your hand.
• You should repeat the experiment a number of times because this will help you estimate masses of different objects.
• With the help of your teacher, measure the mass of a pen, small stone, a small exercise book,your shoes and a bottle top.
• With the help of your teacher, measure and record your mass in your exercise book.
• Compare your mass with the masses of your classmates.
• Record the least and the highest mass in the class.
• Use your imagination to estimate the mass of the teacher.
• Ask the teacher to tell the whole class his/her mass after all pupils have given their estimates.
A butcher at a butchery in Kigali City sells meat. He is 2 m tall, very fat and he puts on the biggest shoe size on market. What does he weigh?
Estimate the mass of the following objects:
(a) mass of a bottle of soda. (b) mass of a pawpaw fruit.
(c) mass of a goat. (d) mass of a bull.
(e) mass of a small car. (f) mass of a lorry.
(g) mass of a knife.
(i) mass of a 10 year old pupil. (j) mass of 10 sheets of paper.
Measuring of Mass
Which unit would you use to measure the mass of the following? A kilogram or a gram?
(a) a tomato (b) an egg (c) a radio
(d) a baby (e) an exercise book (f) a school boy
(g) a chair
Mass is the quantity of matter contained
in a substance. The more the matter, the
greater the mass. A house brick and a
piece of cotton of the same size have
different masses. A house brick has
more mass because it has more
quantity of matter than the cotton.
The standard unit for measuring and expressing mass is kilogram. The
kilogram is represented by ‘kg’.
However, mass can also be expressed in ton, denoted by ‘t’ where,
1 ton = 1 000 kg.
Instruments for measuring mass
The following instruments can be used to measure mass:
Top pan balance, beam balance, triple beam balance, electronic balance, etc.
Which units would you use to measure the following objects:
(a) mass of a pencil. (b) mass of a ball.
(c) mass of a cycle. (d) mass of a TV.
Conversion between units of Mass
Metric prefixes are very useful in converting units of quantities. The main metric prefixes dealt with at this level are the kilo, hector, decal, dici, cent and mill.
From the above table we can see that;
• One kilogram = 1kg = 1000g.
• One hectogram = 1hg = 100g
• One decagram = 1 dag = 10g
• One decigram = 1 dg = 0.1g ( a tenth of a gram)
• One centigram = 1cg = 0.01g = (hundredth part of a gram)
• One milligram = 1mg = 0.001g = (a thousandth of a gram)
Note:
The tonne is equal to one thousand kilograms.
1 tonne (1t) = 1 000kg
1q = 100kg.
Read the following units of mass aloud:
(a) 20 t (b) 250 kg
(c) 400 hg (d) 680 dag
(e) 500 g (f) 230 dg
(g) 100 cg
(i) 45 cg
Example 8.1
Convert the following units of mass:
(a) 2 kg to g. (b) 2 kg to hg.
(c) 10 kg to g. (d) 3 dag to g.
(e) 40 g to mg. (f) 50 dg to cg.
Solution
Conversion table
(a) 2kg = 2 000g
(b) 2kg = 20hg
(c) 10kg = 10 000 000mg
(d) 3 dag = 30g
(e) 40g = 40 000mg
(f) 50dg = 500cg
Example 8.2
Convert the following units of mass:
(a) 2 000 g to kg.
(b) 30 000 g to kg.
(c) 15 000 cg to g.
(d) 200 dag to hg.
Solution
(a) 2000g = 2kg
(b) 30 000g = 30kg
(c) 15 000cg = 150g
(d) 200dag = 20hg
1. Convert the following into grams. (a) 24 kg (b) 8 kg (c) 15 kg (d) 12 kg (e) 1.5 kg
(f) 321 kg (g) 2.8 kg
2. Express the following in kilograms. (a) 15 000 g (b) 2 400 g (c) 7 000 g (d) 500 g
(e) 912 g (f) 1 500 g
Addition and Subtraction of Masses
Example 8.3
(a) Add: 4kg + 3hg = ........kg
(b) Subtract: 29dg – 2.4cg = ........cg
(c) Subtract: 10t – 9 600kg = ........kg
Solution
(a) 4kg + 3hg = 4kg + 0.3kg = 4.3kg
(b) 29dg – 2.4cg = 290cg – 2.4cg = 287.6cg
(c) 10t – 9 600kg = 10 000kg – 9 600kg = 400kg
(a) Add: 3.25 kg and 1.75 kg and express the answer in grams.
(b) Subtract: 5.25 kg from 25.65 kg and write answer in grams.
Application of Mass
In this section, we discuss the problems involving mass in real life situations.
Example 8.4
A school bought shields and cups for winners and runners-ups. There were 2 shields for the overall champions. If the weight of the smaller shield is 2 kg 120 g and bigger shield is 2 kg 865 g, then what is the difference in their masses.
Solution
The weight of the bigger shield = 2 kg 865 g = 2 865 g
The weight of the smaller shield = 2 kg 120 g = 2 120 g
Difference in weight = 2 865 g – 2 120 g = 745 g
1. Convert the following into grams.
(a) 285 kg (b) 19 kg (c) 2.5 kg (d) 196 kg.2. Convert the following into kilograms.
(a) 2126 g (b) 9065 g (c) 850 g
3. There are 20 tins of biscuits in a shop. Each tin weighs 2 kg 500 g. What is the weight of all the tins in kg?
4. A car weighs 3 tons. Express this mass in kg.5. An omnibus (taxi) is licensed to carry passengers with total mass not exceeding 1 140 kg. If 20 people each of mass 60 kg board the omnibus, find whether the omnibus is overloaded.
6. A truck carries 200 bags of cement. The total weight of all the bags in the truck is 10 tons. What is the mass of only 1 bag of cement?
7. Each book in a certain bookshop has a mass of 20 dg. How many of these books do I need to have 1 ton of books?
8. Yesterday, I bought 4 kg of mangoes and ate 2 000 g. How many grams of mangoes are left?
9. The mass of 1 chocolate bar is 100 g. What is the mass of 30 chocolate bars in kg?
10. The mass of 1 toy car is 600 g. What is the total mass of 60 toy cars?
11. Ms Annet packed 24 kg of sugar equally into 8 bags. How many grams of sugar did she pack in each
bag?
12. Peter has a mass of 82 kg. John is 5 kg heavier than Peter. What is the mass of John?
13. John weighs 80 kg, Peter weighs 70 kg and Eric weighs 90 kg. How much do the three people weigh
altogether?
For more online support visit www.kidsnumbers.com
Key Unit Competence
By the end of this unit, a learner should be able to understand area as a two
dimensional (2D) space enclosed by a boundary. The learner should also use
square and land units in solving mathematical problems.
Attitudes and values
Appreciate the need to properly and accurately use different area and land
measurements in daily life situations.
Understanding Area
In the preceding figure, the shaded part (green) is a square of length 1 mm and
width 1 mm. The area of this part is 1 mm × 1 mm = 1 mm2.
Now, consider the big square (red). There are 10 small squares along its length
and 10 squares along its width.
So, the area of the big square = 10 mm × 10 mm = 100 mm2.Important point
If area = 100 mm2, then there are 100 squares each of area 1 mm2.
Refer to the area grid shown:
• What is the area of 1 small square?
• What is the area of the big square?
• Count the number of small squares in
the big square.
In the figure below, each small square has both length and width of 1 cm.
(a) What is the area of one small square?
(b) How many small squares are in the figure?
(c) What is the area of the whole figure?
(d) What is the area of the shaded part?
Units of Area
The standard unit of area is the square metre. It is written as m2. However,
other units for measuring area are km2, hm2, dam2, dm2, cm2 and mm2.
Conversion table for units of area
From the conversion table above, we can see that:
1km2 = 1 000 000 m2
1hm2 = 10 000 m2
1dam2 = 100 m2
1 m2 = 1 000 000 mm2
1dam2 = 10 000 dm2
1cm2 = 100 mm2
The meaning of 1 km2
Consider a square plot of land ABCD such that
BC is 1 km long and CD is also 1 km long as
shown.
Area of the plot ABCD = length of side BC x length of side CD = 1 km x 1 km = 1 km2.
The area of this plot of land is 1 km2, which is read as “one kilometre squared” or “one square kilometre”.
NOTE:
Since 1 km2 = 1 km x 1 km, then we can easily change 1 km2 in to other units of area such as m2.
1 km2 = 1 km x 1 km = 10 hm x 10 hm = 100 hm2.
Read the following areas aloud:
(a) 50 km2 (b) 600 hectares (c) 40 acres
(d) Our school football field has an area of 7 000 m2
(e) My study table is 320 cm2 (f) The size of my wall photo is 300 mm2.
9.2Change the following units of area as instructed:
(a) 1 km2 = _______ m2
(b) 10 m2 = _______ cm2
(c) 25 dam2= __________ m2
Relationship Between Land Measurement and Area
Land area is measured in a larger
unit called the hectare. It is writtenin short as ha. Other units of area include; are, acre and hectare.
• A plot of land measuring 100 m by 100 m has an area of 1 hectare. So,
1 hectare (1 ha)= 100 m × 100 m = 10 000 m2.
• A plot of land measuring 10 m by 10 m has an area of 1 are.So, 1 are = 100 m2.
• 1 ca = 1 m2, i.e. 1 centiare =1 square metre.
• 1 acre = 0.40 hectare and 1 hectare = 2.47 acre.
1ha = 100a
1a = 100 ca
1 ha = 10 000ca
1ca = 1m2
1 hm2 = 1ha, 1dam2 = 1a, 1m2 = 1ca.
Activity
Study the conversion table above. Discuss with your group members and answer the following questions:
• 1a = ..........ca
• 1dm2 = .........ca
• 10a = ............ha
• 20ca = ............cm2
• 5dam2 = ............dm2
• 25m2 = ...............ca
• 25ca = ..............a
• 5 000 m2 = ...............ha
In this class activity, you will find the area of a plot of land of length 10 m and width 5 m by counting the number of square metres in it.
(a) 10 m can not fit into your book, so you will use a scale of 1 cm to represent 1 m. Draw a rectangle of length 10 cm and width 5 cm in your book.
(b) Divide the length into 10 equal parts using vertical lines. Each part should be equal to 1 cm.
(c) Divide the width into 5 equal parts using horizontal lines as shown in
the diagram below:
(d) Count the number of small squares formed by the lines you have
drawn. You can do this by numbering all the small squares.
(e) Each small square represents an area of 1 m2.
(f) Now answer this question: What is the area of the plot of land?
Example 9.1
Convert the following units of area:
(a) 3 m2 to cm2
(b) 5 km2 to dam2
(c) 2.5 hectares to m2
Solution
(a) 3m2 = 30 000cm2
(b) 5km2 = 50 000dam2
(c) 2.5 ha = 25 000m2
9.3Convert the following units:
(a) 5 km2 in m2.
(b) 1 hectare in m2.
(c) 10 are in m2.
Area of Rectangular Piece of Land
Area of a rectangular piece of land = length × width.
mathematically,A = l × w
where A = Area
l = length
w = width
Example 9.2
Find the area of a rectangular piece of land whose length and width are:
(a) 80 m and 20 m. (b) 100 m and 40 m.
Solution
(a) Area = length × width = 80 m × 20 m = 1 600 m2.
(b) Area = 100 m × 40 m = 4 000 m2.
1. The length of a house floor is 10 m and its width is 8 m. What is the area of the floor?
2. The area of the floor of our classroom is 40 m2. How long is the classroom if it is 5 m wide?
Addition and Subtraction of the Area of Land
In groups, do the following:
• Measure the length and width of your classroom.
• Also, measure the length and width of another classroom.
• What is the total area? What is the difference between the areas of two rooms? Write the answers in m2.
In this class activity, you are going to work in groups and find the area of the plot of land shown below. Divide the land in two parts. Find the areas of plot separately. Add the area of the two plots and express in m2.
Example 9.3
1. Add:
(a) 17.25 m2 and 5.05 m2. (b) 57.03 are and 10.07 are.
2. Subtract:
(a) 15.27 are from 24.09 are. (b) 9.23 m2 from 16.67 m2.
Solution
1. Change the following units of area as instructed.
(a) 2 500 cm2 = ________ m2
(b) 8 000 m2= ____________ hm2
(c) 6.5 km2 = ____________ dam2
2. A piece of paper has an area of 6 cm2. If it is 3 cm long, then how wide is the paper?
3. (a) Add 25.32 m2 and 62.28 m2.
(b) Subtract 4.25 ha from 6.75 ha.
4. School A is built on 1 are of land and school B is built on 100 m2 of land. Which of the two schools has bigger area?
5. On a fruit farm, 1 are of land produces 2 tons of oranges. How many kilograms of oranges will be produced by 300 m2 of the same farm land?
6. Kenia has a plot of land measuring 100 m by 100 m. Peter’s land has an area of 3 hectares and Kwame has an area of 200 ares. Who has the largest land? What is the total Land area in hectares?
7. The total land area of our country Rwanda is 26 338 km2. Lake Victoria has an area of 69 484 km2. By how much is Lake Victoria bigger than Rwanda?
Internet Resource www.mathplace.com
Key Unit Competence
The learner should be able to tell, write and convert time appropriately.
Fun Corner
“Who has two hands but can’t wave?”
Ans: Eeeehhhh.....ahahaha....Am I not Mr. Clock? “You are my friend, I am your friend. You can wait for me but I can never wait for you!” Who am I? Ans: Ahahaha...... I am Mr. Time.
Attitudes and values Appreciate the value of time management in daily situations.
Reading and Telling Time Time is a measure of durations of events and the intervals between them.
The short hand of the clock is called the hour hand.
It measures time in hours. If it rotates once round the clock face, then the time taken is 12 hours.
The long hand of the clock is the minute hand and it measures time in minutes.
One full rotation equals 60 minutes.
The thinnest hand is the second hand. Of the three, it rotates the fastest. Its full rotation equals 60 seconds.
There are two common ways of telling time:
(i) Say the hour first and then the minutes e.g.
• 6:25 = It is six twenty-five
• 8:05 = It is eight O-five (O is said like letter O)
• 9:11 = It is nine eleven
• 2:45 = It is two forty-five (ii) Say the minutes first and then the hour e.g.
• 2:16 = It is sixteen past two
• 11:20 = It is twenty past eleven or it is eleven twenty
• 8:30 = It is half past thirty or it is eight thirty
• 8:40 = It is twenty to nine or it is eight forty
• 11:59 = It is one to twelve or it is eleven fifty-nine
• 8:51 = It is nine to nine or it is eight fifty-one.
Quarter past/to and half past 15 minutes is a quarter of 60 minutes and 30 minutes is half of 60 minutes.
• When it is 15 minutes past the hour we use the words “quarter past” e.g. 6:15 = a quarter past six, 9:15 = a quarter past nine etc.
• When it is 15 minutes to the hour, we use the word “a quarter to”, e.g.
10:45 = a quarter to eleven, 4:45 = a quarter to five etc. • When it is 30 minutes past the hour, we use the words “half past” not “half to”. e.g. 7:30 = half past seven, 12:30 = it is half past twelve etc. O’clock
We use o’clock, when there are no minutes. We use o’clock when the time is exact. For example; 2:00 = two o’clock, 11:00 = eleven o’clock etc.
12:00
When it is exactly 12:00, we say it is twelve o’clock. There are 2 twelve o’clocks in a day namely: • Twelve o’clock midday. This is also referred to as noon. • Twelve o’clock midnight.
10.1Read and tell the time in the clocks shown below:
Example 10.1
Read and tell the time shown by the clock faces below:
Solution
(a) The time is 9:40. This may be 9:40 a.m. or 9:40 p.m.
(b) The time is 5:20. It may be 5:20 a.m. early morning or 5:20 p.m. in the evening.
(c) It is one o’clock.
(d) The time is 8:40
(e) The time is 7:40
(f) The time is 2:20
1. Match the digital and analogue clocks which show the same time. You may use an arrow.
2. Read and tell the time indicated by the following clock faces:
3. Draw the hour and minute hands on the following clock faces to show the correct times indicated:
Writing Time Using a.m. and p.m.
Time using a.m. is ante meridian, means before midday.
Time using p.m. is post meridian, means after midday.
a.m. starts at exactly midnight at a time called 12:00 o’clock up to just before midday (12 noon). p.m. starts at midday at a time called 12 noon and ends just before midnight.
Use a manilla card and plastics to form a model clock as shown below. Your teacher will help you. Use it to show the time mentioned below.
Example 10.2
Express 9 o’clock in the morning using a.m. or p.m.
Solution
9 o’clock in the morning is 9:00 a.m.
Example 10.3
What time is a half past 5 o’clock in the evening?
Solution
A half past 5 o’clock in the evening is 5:30 p.m.
1. Write the time using ‘a.m.’ or ‘p.m.’: 10 o’clock in the morning?
2. Express 6 o’clock in the morning as a.m. or p.m.
3. What time is half past 11 o’clock in the morning?
4. Express 2 o’clock in the afternoon using a.m. or p.m.
5. My father left Moroto at a quarter to 7 o’clock in the morning. Write this time using a.m. or p.m.
Units of Time and Their Conversion
The standard unit of time is the second. The second is written in short as ‘s’. Other units used to express time are minute, hour, day, week, fortnight, month, year, decade, century and millennium, where
1 minute = 60 second(s).
1 hour = 60 minutes.
1 day = 24 hours.
1 week = 7 days.
1 fortnight = 2 weeks.
1 month = 4 weeks.
1 year = 12 months (365 days).
1 decade = 10 years.
1 century = 100 years.
1 millennium = 1000 years. • An ordinary (common) year has 365 days and February ends on 28th. • A leap year is a year which has 366 days and its February ends on 29th. • 2004, 2008, 2012 were leap years and the next leap year is 2016. • A full day has both daytime and night time, each of which lasts 12 hours.
In this class activity, you will work in groups and find out the number of seconds that are found in:
(a) 1 minute (b) 1 hour (c) 1 day (d) 1 week
Example 10.4
Change 4 hours to minutes.
Solution
1 hour = 60 minutes 4 hours = (4 × 60) minutes = 240 minutes.
Example 10.5
Change 180 minutes to hours.
Solution
(a) 1 hour (d) 5 hours (g) 6 hours
(b) 3 hours (e) 8 hours
(c) 2 hours (f) 11 hours
The Calendar
A calendar is a chart showing days, weeks and months of a particular year. A calendar helps us to identify days and be informed of special days like Christmas, Independence Day, school term opening and closing days, birthdays, etc.
Example 10.6
Name the days of a week. Solution Days are: Sunday, Monday, Tuesday, Wednesday, Thursday, Friday and Saturday
Example 10.7
Name the months of the year. Solution Months are: January, February, March, April, May, June, July, August, September, October, November, December
In this activity, you will list down all the activities you do in a day right from the time you wake up till you go to sleep.
You will work in small groups with your classmates to finish this activity. Carefully follow the teacher’s instructions.Study the calendar below and answer the following questions:
(a) For which year does the calendar belong?
(b) For which month of the year is the calendar?
(c) What does ‘Mo’ stand for in the calendar?
(d) On what day does the month begin?
(e) On what day does it end?
(f) How many week days does the month have?(g) Ineza played football for 30 minutes on each weekend days of the month. How many seconds did he spend playing football in the whole month?
(i) On what date is the last Sunday of the month?
(j) A certain conference started on the first Monday at 9:00 a.m. and ended at 9:00 a.m. on 29th of the same month. How long (in days) did the conference last?
Form yourselves into groups of 4 learners each. Study the calendar and discuss the questions below:
(a) For which year does this calendar belong?
(b) How many months are found in a year?
(c) Which months have 30 days?
(d) Which months have 31 days?
(e) What is the last month of the year?
(f) In which month were you born?
(g) In which month does Rwanda celebrate independence?
Charles’ mum has three children. The first is called June. The second child is called July. What is the name of the third child?
Using the given calendar, answer the following:
1. How many days are there in the months of January, February (not a leap year) and March?
2. List down all months with 30 days.
3. Write down all months of the year with 31 days.
Problems Involving Time
Example 10.8
Kato, is a brick layer. He makes 120 bricks every day. How many bricks does he make in the month of January, if he works every day?Solution
In one day, he makes 120 bricks. But, January has 31 days. Therefore, in 31 days, he makes = 120 × 31 = 3720 bricks Therefore, he will make 3 720 bricks.
Example 10.9
Dorah is a mother of 3 children. She has a job in town and this job enables her to take care of her children. She goes to town everyday except on Sundays. How many times does she go to town in the month of September, if there are 4 Sundays in September? Solution September has 30 days and there are 4 Sundays. (30 – 4) days = 26 days Therefore, she goes to town 26 times in the month of September.
Example 10.10
It takes Aminyo 2 hours 30 minutes to travel from Kampala to her village in Luwero. If she walks from Luwero town to her village for 1 hour 10 minutes, how much time does she spend moving from Kampala to Luwero town?
Solution
1. In groups, study the following class time table and answer the questions given below:
Gihugu community primary school P4 Time-table
(a) How many periods are there before break on Monday?
(b) On which day is the school’s general assembly held?
(c) How long is planned for P.E. in the whole week?
(d) Which subject is given biggest amount of time and how long?
(e) On Wednesday, the Science teacher came to class at 10:30. He conducted an experiment in our class for 15 minutes. We then used the remaining part of the lesson for doing a test. How long was the test?
(f) Last Friday, the community service was extended by 55 minutes because of rain. At what time did it end?1. Draw the hour and minute hands on the following clock faces to show the correct times indicated:
2. Our school timekeeper always rings the bell at half past 10 o’clock in the morning. Write this time using a.m. or p.m.
3. A match between Arsenal and Everton started at a quarter past 6 o’clock in the evening on Saturday. Write this time in a.m. or p.m.
4. Express the following as minutes:
(a) 80 seconds (d) 150 seconds (g) 240 seconds (j) 360 seconds
(b) 70 seconds (e) 130 seconds
(c) 120 seconds (f) 480 seconds (i) 300 seconds (l) 540 seconds
5. Gloria sold pancakes every day for 2 months as follows:
September, 200 pancakes every day.
October, 100 pancakes every day.
(a) How many pancakes did she sell in September?
(b) How many pancakes did she sell in October?
(c) Find the total number of pancakes Gloria sold in the two months.
6. Bwire started washing clothes at 6:30 a.m. and stopped at 7:45 a.m. How much time did he take to wash
clothes?
For more online support visit www.mathplayground.com
Key Unit Competence
A learner should be able to understand what money is and know its applications
in our daily life.
Attitudes and values
Appreciate the importance of money in daily life situations and show concern
and the need for honesty in spending money.
What is money?
Money is a piece of paper or metal which is legally accepted to be used for
buying goods and services within a country.
A service is something you cannot hold in your hand e.g. education, internet,
advice.
National Bank of Rwanda makes money for use in Rwanda.Characteristics of money:
• It cannot be easily damaged.
• It is very scarce. That means it is hard to get money.
• It is easy to carry.
• It is hard to forge.
• It should be easily divisible into smaller denominations.
• It is accepted by everyone.
Uses of money:
• It is used for buying goods like books, cars, food, mobile phones.• It is used for buying services like education (school fees), medical
treatment, airtime, insurance, etc.
• It is used to keep wealth.
• It is used to pay debts.
Rwandan Currency
The money used in our country (Rwanda) is known as Rwandan francs. It is
usually denoted by ‘’.
In Rwanda, we use both notes (paper money) and coins (metallic money).
Rwandan Currency Coins:
In this class activity, you are provided with 6 different Rwandan coins.
a) Identify the 1, 5, 10, 20, 50 and 100 Frw coins.
b) Discuss with your friends what item can be bought by each coin denomination.Rwandan Currency Notes:
In this class activity you are provided with four Rwandan currency notes.
a) Identify the following notes: 500 Frw, 1 000 Frw, 2 000 Frw and 5 000 Frw.
b) Discuss with your friends what item each currency note can buy in Rwanda.
c) Is there any other currency note whose picture is not shown above?Example 11.1
My mother gave me money to buy a new school uniform. It was 5 150 Frw
in an envelope. In the envelope, there was only one note and two coins.
Identify the note and the coins my mother gave me.
Solution
Total money = 5 150 Frw.
Since there was only one note, it must have been the 5 000 Frw note. There
is no coin for 150 Frw, so the two coins are 100 Frw and 50 Frw.Example 11.2
Paul bought sweets from a shopkeeper. He paid the shopkeeper
550 Frw using only coins of 50 Frw each. How many coins did he gave to
the shopkeeper?Solution
1. What do we call the national currency used in Rwanda?
2. Which Rwanda note has the highest value?
3. A mother gave three notes of 1 000 Frw and five coins of 50 Frw to her son to buy school uniform. What was the total amount of money given by the mother?
4. Three men shared 15 000 Frw equally. How much money did each man get?
Sample Currency Notes from Other Countries
Planning According to Needs and Wants
A need is something you cannot live without. A need is something you must
have for you to survive. Examples of needs are food, shelter, clothing and
medical care.
A want is something you would like to have but you can live without it. It is not
absolutely necessary but it would be a good thing to have. Examples of wants
are music, TV, computer, car, toys, mobile phone, radio, electricity.
Scarcity and Budgeting
Those in towns and cities buy most of their needs and wants including food,
clothes, etc. Those in villages may get food from their gardens and also live in
their own houses for free. However, everyone needs money to buy salt, sugar,
medicine, airtime, soda, bicycle, clothes etc. One of the biggest problems that
we all have is scarcity of money. We don’t always have money for everything
that we need or want.
Therefore, every time we have money, we need to plan carefully for the money
so that we buy only the most important things. This planning is called budgeting.
a) Identify the basic needs from the below list.
b) Plan your budget according to your priorities. How will you spend 5 000 Frw?
Buying and Selling
A person who buys goods or services and sells to others is called a trader or
businessman or businesswoman.
Selling is the act of giving out goods to someone in order to get money.
Buying is the act of giving money to someone in order to get goods that you
need.Cost Price and Selling Price
A shop keeper goes to a factory and buys 1 crate of soda at 4 500 Frw. If he
sells the crate to a customer at 6 500 Frw, then the cost price is 4 500 Frw
and the selling price is 6 500 Frw.Profit and Loss
• If the selling price is more than the cost price, then the shopkeeper makes
a profit. This means he gets extra money and he becomes richer.
• If the selling price is less than the cost price, then the shopkeeper suffers
a loss. This means he loses some money and he becomes poorer.
Example 11.3
Manzi was given 3 000 Frw by his father as pocket money for school
use. He bought a pen at 200 Frw, a book at 500 and geometry set at
900 Frw. How much money is left with him?Solution
Cost of a pen = 200 Frw
Cost of a book = 500 Frw
Cost of a geometry set = 900 Frw
Total amount of money spent by Manzi = (200 + 500 + 900) Frw
= 1 600 Frw
Pocket money given to Manzi by his father = 3 000 Frw
Money left with Manzi = (3 000 – 1 600) Frw
= 1 400 Frw
Example 11.4
A trader went to the market and bought a goat at 20 000 Frw, a sheep at
15 500 Frw and a cock at 5 000 Frw. How much money did he spend in the
market?
Solution
Price of a goat = 20 000 Frw
Price of a sheep = 15 500 Frw
Price of a cock = 5 000 FrwTotal amount of money spent by the trader = (20 000 + 15 500 + 5 000) Frw
= 40 500 Frw
Example 11.5
Christa bought a tray of eggs at 2 000 Frw and sold it to Sandra at
2 200c Frw. Sandra sold it to James at 3 000 Frw and James also sold it to
Alex at 2 500 Frw.
a) What profit did Sandra get?
b) What loss did James bear?
Solution
a) Sandra: Cost price = 2 200 Frw
Selling price = 3 000 Frw
Sandra’s profit = Selling price – Cost price
= (3 000 – 2 200) Frw
= 800 Frw
b) James: Cost price = 3 000 Frw
Selling price = 2 500 Frw
James’ loss = Cost price – Selling price
= (3 000 – 2 500) Frw
= 500 Frw
(a) You will do this activity in pairs, suppose a shopkeeper purchases goods at 2 500 Frw and sells the goods at 3 000 Frw. What extra money does the shopkeeper get? Discuss with your teacher.
(b) Shina went to the market and bought a dress at 5 000 Frw. The dress was too big and so she decided to sell it at 6 000 Frw. Did she gain from the sale of her dress? Discuss with your friend.
1. If the cost of one pen is 500 Frw, then what will be the cost of 4 pens?
2. Keza has 450 Frw. She buys an eraser at 250 Frw and a sweet at
125 Frw. How much does she remain with?
3. One metre of cloth costs 2 600 Frw. Maria went with one 5 000 Frw
and one 2 000 Frw note. She bought two metres of cloth. How much
balance was she given?
4. A plot of land was sold at 30 000 Kenyan shillings. This resulted in a
profit of 4 500 Kenyan shillings. What was the cost price of the land?
5. The bus fare from Kigali to Kampala is 12 000 Frw. How much does
one spend from Kigali to Kampala and back to Kigali?
6. The price of a litre of petrol is 1 200 Frw. A car fuel tank has a storage
capacity of 100 litres. How much does it cost to fill up the tank?
7. MTN charges 60 Frw per minute on calls to Kenya. How long do I
speak to a friend in Kenya if I have 1 200 Frw airtime in my phone?
8. A Samsung TV costs $ 340. How much will be charged for 2 similar
TV sets?
9. The price of a mobile phone is marked as 79 500 Frw. The trader sells
it to me and makes a loss of 3 500 Frw. How much do I pay for the
phone?
10. A refrigerator costs 80 500 Frw. Kamanda wants to buy the refrigerator
and he has 76 900 Frw. How much more money does he need to buy
the refrigerator?
11. A group of students are on a school trip to Gisenyi to tour the Methane
Gas Plant on Lake Kivu. The fare for each student is 6 250 Frw. There
are 40 students in the bus. How much does the conductor of the bus
collect from the students?Internet Resource
For more online support visit www.math-play.com/fractionsKey Unit Competence
A learner should be able to describe and generate number patterns.
Attitudes and values
Appreciate the importance of orderliness in daily life. Learners should be made
to realise the need for orderliness in places such as bus, park, banks, markets,
schools, hospitals and other places in daily life situations.
What is a Number Pattern?A list of numbers which form a sequence is called a number pattern.
Ascending order (increasing order)
The term ascending means ‘going up’. While ordering integers in ascending
order, we arrange integers from smallest to the largest.
For example: 1, 3, 6, 8, 9, 23, 56, 400, .....are arranged in ascending order.
Descending order (decreasing order)
The term descending means coming down. While ordering integers in
descending order, we arrange them from largest to smallest.
For example: 45 000, 340, 34, 20, 6, 2, –3, –567, .....,.....,... are arranged in
descending order.
Importance of orderliness in daily life situations
Arranging numbers in order of size helps us to compare numbers easily from a
given set of numbers.
UNIT 12 Number Patterns
Example 12.1
The list shown below shows the size (area) of some selected countries in
the world. (Source: www.simple.wikipedia.org)
Arrange the land areas of the different countries in
(a) ascending order
(b) descending order
Hence tell the 4th smallest country from this list.
Solution
(a) Ascending order: 0.44, 455, 692, 10 990, 26 338, 26 990, 27 750,
27 830, 32 542.
(b) Descending order: 32 542, 27 830, 27 750, 26 990, 26 338, 10 990, 692,
455, 0.44.
The 4th smallest country has an area of 10 990 km2. It is the 4th from the left
on the ascending list and it is the 4th from the right on the descending list.
So the answer is Jamaica.
Example 12.2
For example, the figure below shows the number pattern –10, –6, –2, ......,
This pattern starts with – 10 and jumps by 4 every time.
Each picture has a number on its face.
Can you workout the number patterns and fill the missing numbers?Example 12.3
Dots like these ones below can also be used to show number patterns.
1 dot 3 dots 6 dots 10 dots 15 dots
The number pattern formed is 1, 3, 6, 10, 15,.......
This pattern starts with 1 and then you add 2, 3, 4, 5 and so on.
Each picture has a number on its face.
Can you workout the number patterns and fill the missing numbers?
Compare your answers with the answers of your classmates from the other groups.
1. Complete the number pattern.
(a) 4, 8, 12, 16, 20, 24, ________
(b) 1, 4, 8, 13, ________
(c) 50, 42, 35, 29, ________
(d) 100, 105, 115, 130, 150, ________
(e) 1, 2, 4, 7, ________, 16
2. The list shown below shows the marks obtained (out of 100) by the students of class 4 in a maths test:
59, 56, 79, 82, 90, 53, 19, 54, 65, 88, 93, 58, 63, 52, 75. Arrange the data in the ascending order and find out the third highest mark.
Progression or Series
There are two types of series or progressions namely; Arithmetic series and Geometric series.
Arithmetic Progression
In an arithmetic progression or arithmetic series, the sequence of numbers are such that the difference between the consecutive terms is constant.
For example: The sequence 3, 5, 7, 9, 11, 13, 15 is an arithmetic progression with a common difference of 2.
ABCDEFWhat you need• 6 empty chalk boxes, labelled A, B, C, D, E and F.• At least 100 bean seeds.• Marker pens.• Manila paper.Activity 12.2
What to do
(i) Place one bean seed in box A.
(ii) Record the number of bean seeds in box A on a manila paper.
(iii) Remove the bean seed in box A and transfer it to box B. Add two more bean seeds in box B. (iv) Count the number of bean seeds in box B and record the number on the manila paper.
(v) Remove all the bean seeds in box B and place them in box C. Add two more bean seeds in box C. (vi) Count the number of bean seeds in box C and record the number on a manila paper.
(vii) By adding two bean seeds each time, repeat the procedures for boxes D, E and F. Record the number of beans in the boxes D, E and F on the manila paper.The numbers on the manila paper forms an Arithmetic progression with a first term of 1 and common difference of 2.Use the bean seeds to form other Arithmetic Progressions.
Rewrite the numbers as shown in your exercise book and write down the next two numbers in the following sequences:
(a) 70, 69, 68, 67, 66, 65, 64, 63,......,......,
(b) 20, 17, 14, 11, 8, 5,......,.......,(
c) 10, 20, 30, 40, 50, 60,.......,.......,
(d) 10, 13, 16, 19, 22, 25,......,.......,
Geometric Progression
A geometric progression is a sequence of numbers in which the next number is obtained by multiplying or dividing the previous number by a fixed number.
Examples of a geometric progression are:
40, 20, 10, 5. The first number is 40. The next number of the sequence is got by dividing by 2 or multiplying by 1/2.
1, 2, 4, 8, 16, 32, 64, 128,....... The first number is 1 and the next number is got by multiplying the previous number by 2.
1. Complete the number pattern:
(a) 4, 7, 12, 19, 28, ________ (b) 20, 23, 28, 35, ________
(c) 1, 7, 18, 34, ________
2. Rewrite the numbers as shown in your exercise book and write down the next two numbers in the following sequences: (a) +12, +10, +8, +6, +4, ...., ...., (b) 0, 40, 80, 120, 160, ..., ...,
3. On a modern art painting, there are 4 green dots in the first row, 12 green dots in the second row, 36 green dots in the third row, 108 green dots in the fourth row, and 324 green dots in the fifth row. If this pattern continues, how many green dots will there be in the sixth row?
4. A restaurant used 2 onions on Friday, 4 onions on Saturday, 6 onions on Sunday, 8 onions on Monday, and 10 onions on Tuesday. If this sequence continues, how many onions will the restaurant use on Wednesday?
5. While at work, Mimi is putting papers into folders. She puts 3 papers in the first folder, 6 papers in the second folder, 9 papers in the third folder, 12 papers in the fourth folder, and 15 papers in the fifth folder. If this sequence continues, how many papers will Mimi put in the sixth folder?
Key Unit Competence
By the end of this unit, a learner should be able to solve missing number
problems involving addition and subtraction.
Attitudes and values
Appreciate the importance of inverse operations when solving missing number
problems and checking answers.
Arithmagon
An arithmagon is a polygon with numbers at its vertices and sum of these
numbers determine the numbers written on its edges. An arithmagon is shown
below:
Here, the numbers 7, 9 and 12 are at the vertices of the Arithmagon and the
numbers 16, 21 and 19 are at the edges.
You can see that 12 + 9 = 21, 12 + 7 = 19 and 9 + 7 = 16.
So, add the numbers at the vertex to get the number on the edge. This rule
applies to this arithmagon only. Other arithmagons have different rules. You
can study them carefully and discover the rules by yourself.
In the arithmagon shown above
A = 19 – 9 = 10
B = 19 – 12 = 7
C = 19 – 17 = 2
You will complete this task in groups of 5
learners.
(a) Find the value of b
(b) Find the value of c
Example 13.1
Fill in the missing number.
Solution
In the arithmagon above, there are 4 numbers at the vertices namely 2, 7, 3
and 12. You can see that 2 + 7 = 9, 7 + 3 = 10, 3 + 12 = 15 and 12 + 2 = 14.
Example 13.2
What number should be there in the box?
(a) 9 +
= 16 (b) 18 – 9 =
(c) 36 ÷
= 9 (d) 5 ×
= 30Solution
Assessment Exercise
1. Study the arithmagons below and complete them by inserting appropriate
numbers:
2. Complete the Arithmagon below by inserting correct numbers.
3. Fill in the missing numbers to make the arithmagon correct.
4. Complete the arithmagons below by filling in the right numbers in the
circles.
5. If a number is multiplied by 6, the answer will be 24. What will be the
value of the number.
6. If ‘X’ is added to 6. The number obtained is 19. What is the value of ‘X’.
If you were running a race and you run past the one in second position, what position would you be in now?
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Key Unit Competence
By the end of this topic, a learner should be able to identify types of lines and
angles and use a protractor to measure angles.Geometry is a branch of mathematics which deals with the measurement of
shape and size of different figures.
Attitudes and values
Appreciate the importance of lines and angles in daily activities, being confident
and accurate when measuring lines and angles.
Lines
A line is a long straight mark which joins any two points. It has no end points. A
line has no thickness and extends forever
Types of lines
• Parallel lines do not meet at all.
• Intersecting lines are lines which meet at a point.
• Perpendicular lines meet at an angle of 90˚.
Activity 14.1
Using a geometry set, draw the following lines on a manila paper.
• 2 horizontal lines • 2 vertical lines
• 2 intersecting lines • 2 perpendicular lines
• 2 oblique lines not cutting each other • 3 parallel lines
Label the lines according to their names.
Measuring the length of a line segment
A pair of compass and a ruler can be used to measure the length of line segment.
Consider the line segment AB shown in the diagram below.
Place one arm of the compass at point A.
Extend the second arm of the compass up to point B.
Transfer the compass on to a ruler as shown below.
The length of the line segment can be read from the ruler.
Example 14.1
(a) Railway track is an example of parallel lines.
(b) Aluminium wires in electric poles are examples of parallel lines.
1. Measure the length of the following line segments by the help of a ruler.
2. In each diagram below, name the pairs of parallel lines that exist.
Angles
An angle is the measure of the amount of turn from one direction to another. An
angle is the measure of the space between two intersecting lines.
The unit for measuring an angle is known as ‘degree’. A degree is written
in short as (°). If you draw a circle and divide the circle into equal parts using
180 diameters, then the amount of turn from one diameter to the other diameter
is 1 degree or 1˚.
Draw a straight line AB
Mark a point ‘X’ at the centre of the line segment AB. Use a protractor to measure
AXB. What will be the size of angle AXB?
Types of angles
a) Acute angle: This is an angle which is greater than 0˚ but less than 90˚.
Examples of acute angles are 30˚ and 72˚.
(b) Right angle: This is an angle which is formed by intersection of two
straight perpendicular lines. The measure of a right angle is 90˚.
(c) Obtuse angle: This is an angle which is greater than 90˚ but less than
180˚. Example of obtuse angle is 135˚.
(d) Straight angle: This is the angle whose value is exactly 180˚. A straight
angle is formed when two straight lines meet end to end and form one
straight line.
(e) Reflex angle: This is an angle whose value is greater than 180˚ but less
than 360˚. Examples of reflex angles are 339˚ and 285˚.
(f) Complementary angles: These are two angles which add up to 90˚. For
example 27˚ and 63˚ are complementary angles as 27˚ + 63˚ = 90˚. We
can say, the complement of 27˚ is 63˚ and the complement of 63˚ is 27˚.
(g) Supplementary angles: These are the angles which add up to 180˚. For
example 90˚ and 90˚ have a sum of 180˚. So, they are supplementary
angles. Other examples of supplementary angles are 121˚ and 59˚, 31˚
and 149˚.
Activity 14.3
1. Do you see any objects that form right angles in your classroom? Name
them.
2. Compare your list with your classmates.
Exercise 14.2
2. Using Set squares and a Ruler, draw perpendicular lines to the given
lines below.
Measuring Angles
The instrument used for measuring an angle is called a protractor.
The above protractor can be used to measure angles from 0° up to 180°.
Angles can be measured either in a clockwise sense or anti-clockwise sense
depending on where the angle is drawn.
• When measuring the angle between two intersecting lines, we place the
protractor so that its centre is at the point of intersection of the two lines.
• We adjust the protractor so that the horizontal line on it runs along one of
the lines.
• We measure the angle by counting the number of degrees from one line
to the next line.
Measure the following angles with the help of Protractor
Example 14.2
Solution
The acute angle between the two lines = 63˚.
Example 14.3
Solution
The angle between the two lines is obtuse. We measure it by using the
clockwise scale of the protractor. The size of this angle is 121˚.
Example 14.4
Solution
The angle between the two lines is measured using the anti-clockwise scale
and its value is 175˚.
Example 14.5
Solution
The angle between the two lines = 130° – 40° = 90°.
Alternatively, the angle = 140° – 50° = 90°.
Activity 14.4
In this activity you will make your own protractor.
You will need the following materials:
• Manila paper
• Pencil
• Markers
• Pair of scissors
• Math set
• Paper glue
• Blackboard protractor
1. Place the Manila paper on a large table and spread it flat on the
table.
2. Place the blackboard protractor in the middle of the Manila paper.
3. Trace the outline of the protractor on the Manila paper using a
pencil.
4. Draw short marks on the manila paper corresponding to the following
angles: 0°, 10°, 20°, 30°, 40°, 50°, 60°, 70°, 80°, 90°, 100°, 110°,
120°, 130°, 140°, 150°, 160°, 170° and 180°.
5. Remove the protractor and mark a point which represents the point
where all lines meet on a protractor.
6. Redraw the lines using a marker.
7. Carefully cut out your Manila protractor using a pair of scissor or
any other useful tool.
8. Cut another piece of Manila paper which is identical to your Manila
protractor. Use paper glue to attach this Manila paper under your
Manila protractor.
You now have a strong Manila protractor.
9. Use your Manila paper to measure different angles in your class
room including the angles at the edges of your table, book etc.
Assessment Exercise
1. Name the following angles:
2. What is the value of the angle shown below?
3. In the figure below, the protractor is used to measure many angles at
once. Find the angle between;
(a) Line A and line F
(b) Line B and line C
(c) Line D and line A
(d) Line B and line E
4. Here are some angles marked using letters a, b, c, d, e and f. Using
a protractor, measure and record the angles in degrees.
5. Measure the length of the following line segments by the help of ruler.
Key Unit Competency
A learner should be able to use geometric properties to classify shapes.
Attitudes and values
Appreciate the use of properties to distinguish shapes and recognise that
special quadrilaterals are a subset of all quadrilaterals.
2D Shapes
2D shapes/figures are two dimensional shapes or figures. These shapes
have only two dimensions, i.e., length and width. They do not have thickness.
Examples of 2D shapes are triangle, square, rectangle, rhombus, parallelogram,
kite, trapezium, pentagon, etc.
Triangle
A triangle is a closed three sided figure. It has
three angles and three sides.
Vertex
A vertex is a point where any two sides meet. The plural of vertex is vertices.
So, a triangle has 3 sides, 3 vertices and 3 angles.
Types of triangles
Triangles can be classified according to their sides or angles.
Triangles in terms of sides
• A scalene triangle is a triangle in which all sides are unequal.
• An isosceles triangle is a triangle in which two sides are equal.
• An equilateral triangle is a triangle in which all sides are equal.
Complete this activity in groups of 5 learners
• You are provided with the following paper/card objects:
Identify the triangles by putting them separately from the other paper
objects.
• How many sides does a triangle have?
• How many angles does a triangle have?
• How many vertices does a triangle have?
Write the names of the triangles in terms of their sides.
(a) Scalene triangle (b) Isosceles triangle (c) Equilateral triangle
Obtuse angled triangle is a triangle in which one angle is obtuse, i.e., more
than 90°.
• Acute angled triangle is a triangle in which all the angles are acute, i.e.,
less than 90˚.
• Right angled triangle is a triangle in which one angle is a right angle (90°).
Write the names of the triangles in terms of their angles.
1. Write the names of the triangles in terms of their sides.
2. Write the names of the triangles in terms of their angles.
Quadrilaterals
A closed plane figure bounded by line-segment is called a polygon. The line
segments are called its sides and the points of intersection of consecutive sides
are called vertices. Line segments joining non-consecutive vertices are called
diagonals.
(a) Look at the following plane figures:
From the above figures we observe that if these figures are folded along a specific line (dotted line). Each figure on the left hand of the dotted line fits exactly on the figure on the right hand side of dotted line. Therefore, if a figure is divided into two coincident parts (mirror images) by a line, then the figure is called symmetrical about that line and the line is called the line of symmetry.
(b) Look at the following figure:
The figure shown above does not have any lines of symmetry or point of symmetry. Yet it seems balanced and has regularity of shape.
Let this figure be rotated through one complete turn (clockwise or anticlockwise) about a point O. These are three occasions when it looks the same as it did in its starting positions these are when it has been rotated through 120°, 240° and 360°. We say that this figure has a rotational symmetry of order 3.
A quadrilateral is a 4 sided, closed, 2 dimensional figure.
Identify the 2D shapes given below by writing their names on each shape:
(a) Move around your classroom, library and the whole school compound.
Identify atleast 10 objects which have two dimensional (2D) shapes.
(b) With reasons, explain to your classmates whether the object is a
square, rectangle, rhombus, trapezium, triangle, etc.
1. Name the triangle in terms of their sides.
2. Which of the following shapes are: triangle, square, rectangle, rhombus, parallelogram, kite, trapezium or pentagon.
3. How many lines of symmetry are there in the following shapes:
4. Write the order of rotational symmetry in the following figures.
By the end of this unit, a learner should be able to use area of rectangle to
determine the area of a triangle and other shapes.
Appreciate that the relationship between area and perimeter.
Area is defined as the amount of space inside the boundary of a flat
(2-Dimensional) object such as triangle, square, circle etc.
The area of a closed figure is the measure of the surface enclosed by it.
In this class activity, you will determine the area of a rectangle by counting the
number of square grids in the rectangle.
1. Draw a rectangle of length 6 cm and width 4 cm on a graph paper.
2. Divide the rectangle into grids by dividing the length into 6 equal parts
and width into 4 equal parts.
3. Count the total number of squares formed in the rectangle, i.e., 24. This
is equal to the area of the rectangle.
4. Now, 6 × 4 = 24 which is equal to the area of rectangle.
5. Thus, we find the area of rectangle = length × width
A rectangular football field measures 100 m by 70 m. What is the area of
the field?
Solution
Area = length x width = 100 m x 70 m = 7 000 m2.
As we have seen in above activity that area of rectangle is length × width. Area
of square is defined as, the number of square units it takes to completely fill a
square. In square all four sides are equal. So, area of square is length × length.
A piece of land is in the form of a square. Its side is 200 m long. What is its
area?
Solution
Area of square= side x side
Area of square= 200 m x 200 m= 40 000 m2
In this class activity, you are going to find the area of a triangle from a rectangle.
1. Draw a rectangle of length 8 cm by 4 cm on a graph paper.
2. Divide this rectangle into thirty two 1 cm2 grids. The area of this rectangle
is 32 cm2.
3. Cut out this rectangle using a pair of scissors along one of its diagonal.
4. This forms two equal triangles.
5. What is the area of each triangle?
6. How does the area of the triangle relate to the area of the rectangle? Now,
let’s look at the formula.
165
In the figure above, PQRS is a rectangle with length 8 cm and width 4 cm. SQR is a triangle with base 8 cm and height 4 cm.
Area of rectangle PQRS = (length × width) = 8 cm × 4 cm
=
32 cm2
The shaded part is a triangle. The area of the triangle is half the area of the rectangle.
The height of a triangle makes an angle of 90˚ with the base.
A triangle has an area of 100 cm2 and a base of 10 cm. What is the height of the triangle?
Calculate the area of the following 2D shapes:
Consider the parallelogram JEFH of base 8 cm, height 4 cm and width 5 cm as shown in the diagram below:
If we cut the parallelogram along the height, we get the shaded part EIJ. We can paste the shaded part to the right hand side of HF.
The new figure formed is EFGI as shown in the figure below:
The figure EFGHI is a rectangle of length 8 cm and width 4 cm.
So, the area of the parallelogram EFHJ = area of rectangle EFGI.
Note: The height of the parallelogram is perpendicular to the base.
Area of the given parallelogram = 8 cm x 4 cm = 32 cm2
Find the area of the parallelogram given below:
Solution
Area of the parallelogram = base × height
Base of the parallelogram = 20 cm
Height of the parallelogram = 8 cm
=20 × 8=160 cm2
In pairs
, • Draw a triangle and a rhombus whose perimeter is 36cm.
• Draw a triangle and a rhombus whose area is 36cm2.
A rhombus is a special type of parallelogram in which all the sides are equal and the diagonals bisect each other at right angle. The area of a rhombus can be given by the formula:
AC and DB are diagonals. AC = 2b and DB = 2a. The two diagonals divide the rhombus into four equal right angled triangles. Each right angled triangle has a base b and height a.
A rhombus has diagonals of lengths 20 cm and 32 cm as shown below. Calculate the area of the rhombus.
Calculate the area of the figures given below.
Use a squared paper to draw a rectangle of length 6 units and height 4 units.
• Use a squared paper to draw a right angled triangle of base 3cm and height 4cm.
• By counting the number of squares, find the area of the rectangle and the area of the triangle.
• Join the two shapes together to form one figure. What is the name of this new figure formed?
• By counting the number of squares, find the area of the new figure formed.
• What have you learnt from this activity? Discuss your finding with your teacher.
A trapezium can be divided into two triangles. This makes it easy for the area to be calculated.
ABCD is a trapezium. Line DB is a diagonal of the trapezium. It divides the trapezium into two triangles namely ABD and DBC.
Area of trapezium = Area of triangle ABD + Area of triangle DBC
Example 16.6
What is the area of the trapezium given below?
The perimeter of a closed figure is the total distance around the given figure.
In this class activity, you will use straws to draw a triangle and you will find its perimeter.
(a) You are provided with ordinary drinking straws.
(b) Cut three straws such that each one has a length of 8 cm.
(c) Join these straws to form a triangle.
(d) What is the perimeter of the triangle?
In this activity, you are going to design a square, rectangle, rhombus, parallelogram and trapezium whose perimeter is 24 cm. You can use straws to design these shapes.
To design a rectangle or parallelogram, you need
– 2 straws each of 4 cm long and
– 2 straws each of 8 cm long
So, perimeter = 2 × 4 + 2 × 8=2 (4 + 8)=2 × 12=24 cm
Therefore, Perimeter of rectangle = 2 (length + width)
Perimeter of parallelogram = 2 (length + width)
Now, to design a square or rhombus, you need
– 4 straws each of 6 cm long
So, Perimeter= 6 + 6 + 6 + 6= 4 × 6 = 24 cm
Therefore, Perimeter of square = 4 × side
Perimeter of rhombus = 4 × side
Since, the sides of a trapezium are different in length.
Therefore, Perimeter of trapezium = sum of all the sides.
Find the perimeter of the rectangle and parallelogram given below:
Find the perimeter of the square and rhombus given below.
Solution
(a) Perimeter of square = 4 × side = 4 × 8 = 32 cm
(b) Perimeter of rhombus = 4 × side = 4 × 6 = 24 cm
Find the perimeter of the given trapezium.
Calculate the perimeter of the shape given below.
Solution
Perimeter = 30 mm + 9 mm + 6 mm + 7 mm + 5 mm + 8 mm + 15 mm+ 14 mm = 94 mm
1. Find the area of the following shapes.
2. Calculate the perimeter of the shapes given below.
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Key Unit Competence
By the end of this unit, you should be able to collect, represent and interpret
data.
Attitudes and values
• Appreciate the importance of data collection in daily life situations.
• Appreciate the importance of interpreting and extracting information from
tables.
• Appreciate the importance of statistics tables and graphs in daily life
situations
Statistics?
It is the branch of mathematics which deals with collection, representation,
interpretation and retrieval of data.
Meaning of data
A collection of information in the form of numerical figures is called data.
Examples of data are:
• Ages of pupils in P4 in our class.
• Heights of all pupils in our school.
• Masses of goats in a market.
• Prices of milk in different markets.
• Number of children born in Rwanda per day.Qualitative and Quantitative data
Quantitative data is the type of data which can be expressed in terms of
numbers. For example:
• Number of cars that pass through a certain town per day.
• Number of pregnant cows in a kraal.
• Marks obtained by pupils in a Math test.
• Ages of people in a market
Qualitative data is a type of data which cannot be expressed in terms of
numbers. In other words, it cannot be quantified. We use other attributes other
than numbers to describe qualitative data.
Examples of qualitative data
• A person’s skin colour; I am black, brown or white.
• A person’s sex; I am either a male or female.
• Someone’s nationality; I am a Rwandan, my friend is a Ugandan and our
neighbour is Congolese.
• My favourite soda; the answer can be coca cola, fanta orange or pepsi.
Ways of collecting data
There are many ways of collecting data which may include the following:
1. Questionnaire
A questionnaire is a written set of questions that are given to people in order to
collect facts or opinions (data) about something.
2. Census
A census is a study that obtains data from every member of a population. A
census is very expensive because it is very hard to reach everybody. A national
population is carried out to determine the number of people in a country.
In Rwanda, the last Housing and Population census was carried out in August
2012.
3. Sample survey
A sample survey is a study that obtains data from a very small group of a
population. The result of the sample survey is used to make conclusion for the
whole population.
4. Observation
This is where the person collecting the data makes observations and compiles
his/her data accordingly. For instance, a person can simply observe and count
the number of girls in a class without necessarily talking to anyone.5. Interview
In this method of collecting data, people are asked some questions directly.
Presentation of Data
Presentation of data is a method of displaying data in a very simple way so that
the data collected can be easily understood. Data can be displayed in many
forms namely:
• Table
• Bar graph
• Pie chart
• Line graph
• Pictograph
However, at this level, we shall look at table and bar graph only.
Table
This is a very simple way of presenting data. The data is presented in columns
and rows.
Example 17.1
The data below shows the results of football matches by three teams in a
football league:
Kanombe United: Win = 6, Draw = 3, Loss = 1
Kigali Stars: Win = 6, Draw = 0, Loss = 4
Gisenyi Boys: Win = 4, Draw = 2, Loss = 4
(a) Present the information on a table.
(b) If a win is worth 3 points, a draw 1 point and a loss is worth 0 point,
which team has got the highest points?
Solution
(a) Let us arrange our data in a table like the one below:
(b) Total points:
Kanombe United = (6 × 3) + (3 × 1) + (1 × 0) = 18 + 3 + 0 = 21 points.
Kigali Stars = (6 × 3) + (0 × 1) + (4 × 0) = 18 + 0 + 0 = 18 points.Gisenyi Boys = (4 × 3) + (2 × 1) + (4 × 0) = 12 + 2 + 0 = 14 points.
Kanombe United has the highest points = 21 points.
In this class activity, each pupil will ask the age of all the pupils of the class and represent this data in a table.
Example 17.2
The data below shows the marks obtained (out of 100) by some pupils in a
Maths test:
56 56 76 23 09 89 90 43 23 12 64 88
43 54 76 23 22 43 12 54 29 66 74 50
(a) Represent this data in a table.
(b) How many pupils did appear for the test?
(c) How many pupils scored 76% marks?
(d) How many pupils got less than 50% marks?
(e) What was the highest marks scored?
Solution
(a)
(b) Total number of pupils who appeared for the test = 24 pupils.
(c) 2 pupils scored 76% marks.
(d) Number of students who got less than 50%
= 1 + 2 + 1 + 3 + 1 + 3 = 11 pupils.
(e) The highest mark scored was 90%.
20 pupils are asked for their favourite colour. The answers are as follows:
Red, Red, Blue, Black, Yellow, Green, Green, Black, Pink, Pink, Pink, Blue, Blue, Yellow, Green, Red, Black, Red, Red, Pink
(a) Represent this data in a table.
(b) Which colour is liked by the most number of pupils?
(c) Which colour is liked by the least number of pupils?
Bar graph
A bar graph is drawn by representing the data on two perpendicular axis. The
vertical axis is called the y-axis and the horizontal axis is called the x-axis.
Example 17.3
The data below shows the number of boys from P1 to P5 in Gisozi GS
Primary School.
Represent this data on a bar graph.
Solution
The vertical axis shows the number of boys and the horizontal axis represents
the classes.
From the graph we can see that;
• P1 has 10 boys, P2 has 15 boys, P3 has 20 boys, P4 has 5 boys and P5
has 15 boys.
• P3 has the most number of boys and P4 has the least number of boys.
• Total number of boys from P1 to P5 = 10 +15 + 20 + 5 + 15 = 65 boys.
Ask each pupil about their favourite fruit. Collect the data in a table. Represent
this data by a bar graph.Example 17.4
A survey was carried out in a certain school to find out the means of transport
used by pupils to go to school. It was found out that each pupil go to school by
bus, on foot or by bicycle. The data is shown in the bar graph below:
(a) What quantity is represented on the vertical axis?
(b) What quantity is represented on the horizontal axis?
(c) How many pupils go to school by bus in P4?
(d) How many pupils go to school by bus in P3?
(e) How many pupils go to school by bicycle in the whole school?
(f) Which class has the largest number of pupils?
Solution
(a) “Number of pupils” is represented on the vertical axis.
(b) “Classes” is represented on the horizontal axis.(c) 4 pupils go to school by bus in P4.
(d) There is no bar for the bus in P3. This means no pupil in P3 goes to
school by bus.
(e) Total number of pupils who go to school by bicycle = 2 + 4 + 7 + 5 + 3
+ 4 = 25 pupils.
(f) There are 16 pupils in P4. So P4 has the largest number of pupils in
the school.
1. The birth rate per thousand of five countries over a period of time is shown below:
Represent above data in a bar graph.
2. The table given below shows the monthly expenditure (in percentage) on various items, in a family.
Represent the above data in a bar graph.
3. The graph below shows the salary of five ladies who work in a certain factory. Use it to answer the questions that follow.
(a) How much money does each lady get?
(b) Find the sum of Safina’s and Ann’s salary.
(c) Find the average of Julie’s, Safina’s and Rose’s salary.
Key Unit Competence
By the end of this unit, a learner should be able to play games based on the
probability.
Attitudes and values
Appreciate the importance of following rules and taking turns when playing
games of cards and coins, throwing dice, snakes and ladders and bingo with
probability.
Meaning of Probability
Probability is a branch of Mathematics which deals with the possibility of
something happening. The possibility of something happening is also called
the chance or likelihood of something happening.
Games such as tossing a coin, cards, throwing dice, snakes and ladders and
bingo etc. are played based on probability.
Tossing a coin
• Pair up with one of your classmates.
• Together with your friend, you will decide which side of the coin is the head and which one is the tail. Confirm with the teacher which side is the head or tail.
Each one of you will draw a table like the one shown below in your book.
You will toss the coin, allow it to fall on the floor and note the side facing
up. If the head is facing up in the 1st toss, then write H in the table.
Otherwise, write T for a tail facing up.
• Repeat the experiment 9 more times and record your results as honestly
as possible in the table.
• Let your friend also repeat what you have done and help him/her record
his/her results in a table.
• Compare your results and see who has more heads than the other.
Tossing 3 coins at once
• In this activity, you will pair up with a classmate.
• Carefully observe the coin faces and decide which the heads are.
• You will be required to toss all the three coins at the same time and note the sides of the coin facing up. It is good to decide on a winning combination before starting the game. For instance, the winning combination may be getting 2 heads and 1 tail. If you get anything else, then you have lost.
• Toss the three coins at once and note the sides facing up. Record your result as a win or a loss.
• Give the three coins to your friend and let him/her toss.
• Repeat the experiment several times and see who gets 10 wins first.
- In this class activity, you will play a game of cards and discover how lucky you
- are!!!!
(a) Open the pack of playing cards. Count the total number of cards and
record your answer.
(b) Group the cards of the same type together. You should have 5 groups of
cards namely; hearts, spades, clubs, diamonds and jokers.
(c) Count and record the number of each suit of cards.
(d) How many picture cards are there altogether?
(e) Shuffle the pack of playing cards very well. Deal 5 cards to your neighbor.
(f) Open the cards so that everybody can see. Count the number of aces he/
she has got and record the number down.
(g) Repeat the shuffling and dealing for everybody in the group.
aces each pupil has got.
(i) The pupil who gets 10 aces first is the winner of this game.
(j) Is the game fair? If you think it is unfair, discuss how you could make it
fair for everyone.
1. Discuss the following questions: (Refer Activity 18.1)
(a) Was the game fair?
(b) Are there some winning strategies for getting a head up?
(c) Some people believe that girls will get more “heads” up than “tails”. Do you agree?
(d) Some people think that tossing the coin first makes you get “tails” only. Is this true?
2. Discuss the following questions: (Refer Activity 18.2)
(a) Were we very sincere while doing the experiment? (
b) Is it possible to get 2 heads and a tail?
(c) Some people believe that small coins show heads only and big coins show tails only. Do you agree with this?
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- GLOSSARY
- Abacus: a table or frame used for performing arithmetical calculations.
- Acre: a unit of land measure equivalent to 4046 square metres.
- Addends: the numbers being added together.
- Angle: circular measure of the space between two intersecting lines.
- Acute angle: Angle less than 90˚.
- A.M. : Ante Meridian, time before midday.
- Architect: Stet person who designs and constructs buildings.
- Area: measure of the 2D space surrounded by lines.
- Arithmagon: a polygon with numbers at its vertices which determine the numbers written on its edges.
- Arithmetic: branch of mathematics that deals with calculations
- Calculator: instrument used to help adding, subtracting, multiplying and dividing numbers.
- Calendar: an arrangement of dates, days and months of the year.
- Capacity: the amount of liquid a given container can carry.
- Circle: closed figure with all points at equal distance from the centre.
- Circumference: distance around a circle.
- Cube: closed figure with six equal faces.
- Coin: metallic money in the form of a small disc
- Compass: device for telling the direction.
- Data: any information collected for statistics.
- Denominator: the lower number in a fraction.
- Diagonal: a line (which is not a side) joining any two vertices of a polygon.
- Diameter: distance from one point on a circumference to the other through the centre. It is twice the
radius of a circle.
- Digit: a numeral or number.
- Edge: the boundary line of a surface.
- Even number: a number which is divisible by 2.
- Equivalent fractions: two fractions which have the same numerical value.
- Estimate: to approximate something.
- Face: any of the flat surface of a solid figure.
- Factors: any two numbers multiplied together to form another number.
- Fraction: ratio of two numbers; the numerator and the denominator.
- Gram: a unit of mass equivalent to 0.001 kg.
- Graph: a diagram displaying data showing the relation between two quantities.
- G.C.F. : Greatest Common Factor.
- H.C.F. : Highest Common Factor.
- Intersect: for lines, to meet at a point.
- L.C.M.: Lowest Common Multiple.
- Line: a straight path through two or more points.
- Litre: the unit of capacity equivalent to 1000 cm3.
- Multiple: a number that may be divided by another number with no remainder.
- Numerator: the lower number of a fraction.
- Oblique line: a line which is neither horizontal nor vertical. It is a sloping/slanting line.
- Pair of compasses: a tool consisting of two arms used to draw circles.
- Parallel lines: these are lines which do not meet at all.
- Pentagon: a 5-sided closed figure.
- Perimeter: the distance around a given figure
- Perpendicular lines: two or more lines which intersect at an angle of 90°.
- Plane: a flat surface.
- Point: a small dot used to represent the location of something.
- Polygon: a closed figure bounded by straight or curved edges.
- Prime number: a number with only two factors. One factor being 1 and the other one is itself.
- Probability: this is the chance or likelihood of something happening.
- Protractor: This is a geometrical instrument used for measuring angles.
- Quadrilateral: This is any 4-sided closed figure.
- Radius: this is the distance from the centre of a circle to any point on the circumference of the
circle It is half the diameter.
- Range: this is the difference between the largest and smallest numbers in a data.
- Ratio: it refers to the division of two numbers.
- Remainder: a number which is left when two numbers are divided.
- Right angle: an angle which is equal to 90°.
- Square: a closed 4-sided figure with all sides equal.
- Symmetrical shape: a shape that can be separated into two exactly equal parts which can overlap.
- Tossing a coin: throwing a coin in air so that it falls with one side facing upwards.
- Triangle: a closed figure with three sides, three angles and three vertices.
- Vertex: a point on a polygon where two lines meet.
- Volume: the space occupied by an object.
- X-axis: the horizontal line on the Cartesian plane.
- Y-axis: the vertical line on the Cartesian plane.
- In this class activity, you will play a game of cards and discover how lucky you
MATHEMATICS: P4
TOPIC AREA: ALGEBRA
- ASSESSMENT File Uploaded 24/04/20, 14:20